Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India,[1] Persia (Iran)[2], China, Germany, and Italy.[3]

The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number in the first (or any other) row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

PascalTriangleAnimated2
In Pascal's triangle, each number is the sum of the two numbers directly above it.

Formula

A diagram that shows Pascal's triangle with rows 0 through 7.

The entry in the nth row and kth column of Pascal's triangle is denoted . For example, the unique nonzero entry in the topmost row is . With this notation, the construction of the previous paragraph may be written as follows:

,

for any non-negative integer n and any integer k between 0 and n, inclusive.[4] This recurrence for the binomial coefficients is known as Pascal's rule.

Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices.

History

Yanghui triangle
Yang Hui's triangle, as depicted by the Chinese using rod numerals, appears in a mathematical work by Zhu Shijie, dated 1303. The title reads "The Old Method Chart of the Seven Multiplying Squares" (Chinese: 古法七乘方圖; the fourth character 椉 in the image title is archaic).
TrianguloPascal
Blaise Pascal's version of the triangle

The pattern of numbers that forms Pascal's triangle was known well before Pascal's time. Pascal innovated many previously unattested uses of the triangle's numbers, uses he described comprehensively in what is perhaps the earliest known mathematical treatise to be specially devoted to the triangle, his Traité du triangle arithmétique (1654; published 1665). Centuries before, discussion of the numbers had arisen in the context of Indian studies of combinatorics and of binomial numbers and Greeks' study of figurate numbers.[5]

From later commentary, it appears that the binomial coefficients and the additive formula for generating them, , were known to Pingala in or before the 2nd century BC.[6][7] While Pingala's work only survives in fragments, the commentator Varāhamihira, around 505, gave a clear description of the additive formula,[7] and a more detailed explanation of the same rule was given by Halayudha, around 975. Halayudha also explained obscure references to Meru-prastaara, the Staircase of Mount Meru, giving the first surviving description of the arrangement of these numbers into a triangle.[7][8] In approximately 850, the Jain mathematician Mahāvīra gave a different formula for the binomial coefficients, using multiplication, equivalent to the modern formula .[7] In 1068, four columns of the first sixteen rows were given by the mathematician Bhattotpala, who was the first recorded mathematician to equate the additive and multiplicative formulas for these numbers.[7]

At around the same time, the Persian mathematician Al-Karaji (953–1029) wrote a now lost book which contained the first description of Pascal's triangle.[9][10][11] It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám (1048–1131); thus the triangle is also referred to as the Khayyam triangle in Iran.[12] Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.[13]

Pascal's triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian (1010–1070). In the 13th century, Yang Hui (1238–1298) presented the triangle and hence it is still called Yang Hui's triangle (杨辉三角; 楊輝三角) in China.[14]

In the west, the binomial coefficients were calculated by Gersonides in the early 14th century, using the multiplicative formula for them.[7] Petrus Apianus (1495–1552) published the full triangle on the frontispiece of his book on business calculations in 1527. This is the first record of the triangle in Europe.[15] Michael Stifel published a portion of the triangle (from the second to the middle column in each row) in 1544, describing it as a table of figurate numbers.[7] In Italy, Pascal's triangle is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolò Fontana Tartaglia (1500–1577), who published six rows of the triangle in 1556.[7] Gerolamo Cardano, also, published the triangle as well as the additive and multiplicative rules for constructing it in 1570.[7]

Pascal's Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published in 1655. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) who called it "Table de M. Pascal pour les combinaisons" (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it "Triangulum Arithmeticum PASCALIANUM" (Latin: Pascal's Arithmetic Triangle), which became the modern Western name.[16]

Binomial expansions

Binomial theorem visualisation
Visualisation of binomial expansion up to the 4th power

Pascal's triangle determines the coefficients which arise in binomial expansions. For an example, consider the expansion

(x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2.

Notice the coefficients are the numbers in row two of Pascal's triangle: 1, 2, 1. In general, when a binomial like x + y is raised to a positive integer power we have:

(x + y)n = a0xn + a1xn−1y + a2xn−2y2 + ... + an−1xyn−1 + anyn,

where the coefficients ai in this expansion are precisely the numbers on row n of Pascal's triangle. In other words,

This is the binomial theorem.

Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of yn in these binomial expansions, while the next diagonal corresponds to the coefficient of xyn−1 and so on.

To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of (x + 1)n+1 in terms of the corresponding coefficients of (x + 1)n (setting y = 1 for simplicity). Suppose then that

Now

Six rows Pascal's triangle as binomial coefficients

The two summations can be reorganized as follows:

(because of how raising a polynomial to a power works, a0 = an = 1).

We now have an expression for the polynomial (x + 1)n+1 in terms of the coefficients of (x + 1)n (these are the ais), which is what we need if we want to express a line in terms of the line above it. Recall that all the terms in a diagonal going from the upper-left to the lower-right correspond to the same power of x, and that the a-terms are the coefficients of the polynomial (x + 1)n, and we are determining the coefficients of (x + 1)n+1. Now, for any given i not 0 or n + 1, the coefficient of the xi term in the polynomial (x + 1)n+1 is equal to ai−1 + ai. This is indeed the simple rule for constructing Pascal's triangle row-by-row.

It is not difficult to turn this argument into a proof (by mathematical induction) of the binomial theorem. Since (a + b)n = bn(a/b + 1)n, the coefficients are identical in the expansion of the general case.

An interesting consequence of the binomial theorem is obtained by setting both variables x and y equal to one. In this case, we know that (1 + 1)n = 2n, and so

In other words, the sum of the entries in the nth row of Pascal's triangle is the nth power of 2. This is equivalent to the statement that the number of subsets (the cardinality of the power set) of an n-element set is , as can be seen by observing that the number of subsets is the sum of the number of combinations of each of the possible lengths, which range from zero through to n.

Combinations

A second useful application of Pascal's triangle is in the calculation of combinations. For example, the number of combinations of n things taken k at a time (called n choose k) can be found by the equation

But this is also the formula for a cell of Pascal's triangle. Rather than performing the calculation, one can simply look up the appropriate entry in the triangle. Provided we have the first row and the first entry in a row numbered 0, the answer will be located at entry k in row n. For example, suppose a basketball team has 10 players and wants to know how many ways there are of selecting 8. The answer is entry 8 in row 10, which is 45; that is, 10 choose 8 is 45.

Relation to binomial distribution and convolutions

When divided by 2n, the nth row of Pascal's triangle becomes the binomial distribution in the symmetric case where p = 1/2. By the central limit theorem, this distribution approaches the normal distribution as n increases. This can also be seen by applying Stirling's formula to the factorials involved in the formula for combinations.

This is related to the operation of discrete convolution in two ways. First, polynomial multiplication exactly corresponds to discrete convolution, so that repeatedly convolving the sequence {..., 0, 0, 1, 1, 0, 0, ...} with itself corresponds to taking powers of 1 + x, and hence to generating the rows of the triangle. Second, repeatedly convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for a sum of n independent copies of that variable; this is exactly the situation to which the central limit theorem applies, and hence leads to the normal distribution in the limit.

Patterns and properties

Pascal's triangle has many properties and contains many patterns of numbers.

Pascal's Triangle animated binary rows
Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent ones and the dark pixels are zeroes.

Rows

  • The sum of the elements of a single row is twice the sum of the row preceding it. For example, row 0 (the topmost row) has a value of 1, row 1 has a value of 2, row 2 has a value of 4, and so forth. This is because every item in a row produces two items in the next row: one left and one right. The sum of the elements of row n is equal to 2n.
  • Taking the product of the elements in each row, the sequence of products (sequence A001142 in the OEIS) is related to the base of the natural logarithm, e.[17][18] Specifically, define the sequence sn as follows:
Then, the ratio of successive row products is
and the ratio of these ratios is
The right-hand side of the above equation takes the form of the limit definition of e
  • Pi can be found in Pascal's triangle through the Nilakantha infinite series.[19]
  • The value of a row, if each entry is considered a decimal place (and numbers larger than 9 carried over accordingly), is a power of 11 ( 11n, for row n). Thus, in row 2, ⟨1, 2, 1⟩ becomes 112, while ⟨1, 5, 10, 10, 5, 1⟩ in row five becomes (after carrying) 161,051, which is 115. This property is explained by setting x = 10 in the binomial expansion of (x + 1)n, and adjusting values to the decimal system. But x can be chosen to allow rows to represent values in any base.
    • In base 3: 1 2 13 = 42 (16)
    • ⟨1, 3, 3, 1⟩ → 2 1 0 13 = 43 (64)
    • In base 9: 1 2 19 = 102 (100)
    •               1 3 3 19 = 103 (1000)
    • ⟨1, 5, 10, 10, 5, 1⟩ → 1 6 2 1 5 19 = 105 (100000)
    In particular (see previous property), for x = 1 place value remains constant (1place=1). Thus entries can simply be added in interpreting the value of a row.
  • Some of the numbers in Pascal's triangle correlate to numbers in Lozanić's triangle.
  • The sum of the squares of the elements of row n equals the middle element of row 2n. For example, 12 + 42 + 62 + 42 + 12 = 70. In general form:
  • On any row n, where n is even, the middle term minus the term two spots to the left equals a Catalan number, specifically the (n/2 + 1)th Catalan number. For example: on row 4, 6 − 1 = 5, which is the 3rd Catalan number, and 4/2 + 1 = 3.
  • In a row p where p is a prime number, all the terms in that row except the 1s are multiples of p. This can be proven easily, since if , then p has no factors save for 1 and itself. Every entry in the triangle is an integer, so therefore by definition and are factors of . However, there is no possible way p itself can show up in the denominator, so therefore p (or some multiple of it) must be left in the numerator, making the entire entry a multiple of p.
  • Parity: To count odd terms in row n, convert n to binary. Let x be the number of 1s in the binary representation. Then the number of odd terms will be 2x. These numbers are the values in Gould's sequence.[20]
  • Every entry in row 2n-1, n ≥ 0, is odd.[21]
  • Polarity: When the elements of a row of Pascal's triangle are added and subtracted together sequentially, every row with a middle number, meaning rows that have an odd number of integers, gives 0 as the result. As examples, row 4 is 1 4 6 4 1, so the formula would be 6 – (4+4) + (1+1) = 0; and row 6 is 1 6 15 20 15 6 1, so the formula would be 20 – (15+15) + (6+6) – (1+1) = 0. So every even row of the Pascal triangle equals 0 when you take the middle number, then subtract the integers directly next to the center, then add the next integers, then subtract, so on and so forth until you reach the end of the row.

Diagonals

Pascal triangle simplex numbers
Derivation of simplex numbers from a left-justified Pascal's triangle

The diagonals of Pascal's triangle contain the figurate numbers of simplices:

The symmetry of the triangle implies that the nth d-dimensional number is equal to the dth n-dimensional number.

An alternative formula that does not involve recursion is as follows:

where n(d) is the rising factorial.

The geometric meaning of a function Pd is: Pd(1) = 1 for all d. Construct a d-dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to Pd(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find Pd(x), have a total of x dots composing the target shape. Pd(x) then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore P0(x) = 1 and P1(x) = x, which is the sequence of natural numbers. The number of dots in each layer corresponds to Pd − 1(x).

Calculating a row or diagonal by itself

There are simple algorithms to compute all the elements in a row or diagonal without computing other elements or factorials.

To compute row with the elements , , ..., , begin with . For each subsequent element, the value is determined by multiplying the previous value by a fraction with slowly changing numerator and denominator:

For example, to calculate row 5, the fractions are  and , and hence the elements are  ,   ,   , etc. (The remaining elements are most easily obtained by symmetry.)

To compute the diagonal containing the elements , , , ..., we again begin with and obtain subsequent elements by multiplication by certain fractions:

For example, to calculate the diagonal beginning at , the fractions are  , ..., and the elements are ,   ,   , etc. By symmetry, these elements are equal to , , , etc.

Overall patterns and properties

Sierpinski Pascal triangle
A level-4 approximation to a Sierpinski triangle obtained by shading the first 32 rows of a Pascal triangle white if the binomial coefficient is even and black if it is odd.
  • The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the fractal called the Sierpinski triangle. This resemblance becomes more and more accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern is the Sierpinski triangle, assuming a fixed perimeter.[22] More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other similar patterns.
a4 white rook b4 one c4 one d4 one
a3 one b3 two c3 three d3 four
a2 one b2 three c2 six 10
a1 one b1 four 10 20
Pascal's triangle overlaid on a grid gives the number of distinct paths to each square, assuming only rightward and downward movements are considered.
  • In a triangular portion of a grid (as in the images below), the number of shortest grid paths from a given node to the top node of the triangle is the corresponding entry in Pascal's triangle. On a Plinko game board shaped like a triangle, this distribution should give the probabilities of winning the various prizes.
Pascal's Triangle 4 paths
  • If the rows of Pascal's triangle are left-justified, the diagonal bands (colour-coded below) sum to the Fibonacci numbers.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1

Construction as matrix exponential

Binomial matrix as matrix exponential. All the dots represent 0.

Due to its simple construction by factorials, a very basic representation of Pascal's triangle in terms of the matrix exponential can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, … on its subdiagonal and zero everywhere else.

Connections to geometry of polytopes

Pascal's triangle can be used as a lookup table for the number of elements (such as edges and corners) within a polytope (such as a triangle, a tetrahedron, a square and a cube).

Number of elements of simplices

Let's begin by considering the 3rd line of Pascal's triangle, with values 1, 3, 3, 1. A 2-dimensional triangle has one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements (vertices, or corners). The meaning of the final number (1) is more difficult to explain (but see below). Continuing with our example, a tetrahedron has one 3-dimensional element (itself), four 2-dimensional elements (faces), six 1-dimensional elements (edges), and four 0-dimensional elements (vertices). Adding the final 1 again, these values correspond to the 4th row of the triangle (1, 4, 6, 4, 1). Line 1 corresponds to a point, and Line 2 corresponds to a line segment (dyad). This pattern continues to arbitrarily high-dimensioned hyper-tetrahedrons (known as simplices).

To understand why this pattern exists, one must first understand that the process of building an n-simplex from an (n − 1)-simplex consists of simply adding a new vertex to the latter, positioned such that this new vertex lies outside of the space of the original simplex, and connecting it to all original vertices. As an example, consider the case of building a tetrahedron from a triangle, the latter of whose elements are enumerated by row 3 of Pascal's triangle: 1 face, 3 edges, and 3 vertices (the meaning of the final 1 will be explained shortly). To build a tetrahedron from a triangle, we position a new vertex above the plane of the triangle and connect this vertex to all three vertices of the original triangle.

The number of a given dimensional element in the tetrahedron is now the sum of two numbers: first the number of that element found in the original triangle, plus the number of new elements, each of which is built upon elements of one fewer dimension from the original triangle. Thus, in the tetrahedron, the number of cells (polyhedral elements) is 0 + 1 = 1; the number of faces is 1 + 3 = 4; the number of edges is 3 + 3 = 6; the number of new vertices is 3 + 1 = 4. This process of summing the number of elements of a given dimension to those of one fewer dimension to arrive at the number of the former found in the next higher simplex is equivalent to the process of summing two adjacent numbers in a row of Pascal's triangle to yield the number below. Thus, the meaning of the final number (1) in a row of Pascal's triangle becomes understood as representing the new vertex that is to be added to the simplex represented by that row to yield the next higher simplex represented by the next row. This new vertex is joined to every element in the original simplex to yield a new element of one higher dimension in the new simplex, and this is the origin of the pattern found to be identical to that seen in Pascal's triangle. The "extra" 1 in a row can be thought of as the -1 simplex, the unique center of the simplex, which ever gives rise to a new vertex and a new dimension, yielding a new simplex with a new center.

Number of elements of hypercubes

A similar pattern is observed relating to squares, as opposed to triangles. To find the pattern, one must construct an analog to Pascal's triangle, whose entries are the coefficients of (x + 2)Row Number, instead of (x + 1)Row Number. There are a couple ways to do this. The simpler is to begin with Row 0 = 1 and Row 1 = 1, 2. Proceed to construct the analog triangles according to the following rule:

That is, choose a pair of numbers according to the rules of Pascal's triangle, but double the one on the left before adding. This results in:

The other way of manufacturing this triangle is to start with Pascal's triangle and multiply each entry by 2k, where k is the position in the row of the given number. For example, the 2nd value in row 4 of Pascal's triangle is 6 (the slope of 1s corresponds to the zeroth entry in each row). To get the value that resides in the corresponding position in the analog triangle, multiply 6 by 2Position Number = 6 × 22 = 6 × 4 = 24. Now that the analog triangle has been constructed, the number of elements of any dimension that compose an arbitrarily dimensioned cube (called a hypercube) can be read from the table in a way analogous to Pascal's triangle. For example, the number of 2-dimensional elements in a 2-dimensional cube (a square) is one, the number of 1-dimensional elements (sides, or lines) is 4, and the number of 0-dimensional elements (points, or vertices) is 4. This matches the 2nd row of the table (1, 4, 4). A cube has 1 cube, 6 faces, 12 edges, and 8 vertices, which corresponds to the next line of the analog triangle (1, 6, 12, 8). This pattern continues indefinitely.

To understand why this pattern exists, first recognize that the construction of an n-cube from an (n − 1)-cube is done by simply duplicating the original figure and displacing it some distance (for a regular n-cube, the edge length) orthogonal to the space of the original figure, then connecting each vertex of the new figure to its corresponding vertex of the original. This initial duplication process is the reason why, to enumerate the dimensional elements of an n-cube, one must double the first of a pair of numbers in a row of this analog of Pascal's triangle before summing to yield the number below. The initial doubling thus yields the number of "original" elements to be found in the next higher n-cube and, as before, new elements are built upon those of one fewer dimension (edges upon vertices, faces upon edges, etc.). Again, the last number of a row represents the number of new vertices to be added to generate the next higher n-cube.

In this triangle, the sum of the elements of row m is equal to 3m. Again, to use the elements of row 4 as an example: 1 + 8 + 24 + 32 + 16 = 81, which is equal to .

Counting vertices in a cube by distance

Each row of Pascal's triangle gives the number of vertices at each distance from a fixed vertex in an n-dimensional cube. For example, in three dimensions, the third row (1 3 3 1) corresponds to the usual three-dimensional cube: fixing a vertex V, there is one vertex at distance 0 from V (that is, V itself), three vertices at distance 1, three vertices at distance 2 and one vertex at distance 3 (the vertex opposite V). The second row corresponds to a square, while larger-numbered rows correspond to hypercubes in each dimension.

Fourier transform of sin(x)n+1/x

As stated previously, the coefficients of (x + 1)n are the nth row of the triangle. Now the coefficients of (x − 1)n are the same, except that the sign alternates from +1 to −1 and back again. After suitable normalization, the same pattern of numbers occurs in the Fourier transform of sin(x)n+1/x. More precisely: if n is even, take the real part of the transform, and if n is odd, take the imaginary part. Then the result is a step function, whose values (suitably normalized) are given by the nth row of the triangle with alternating signs.[23] For example, the values of the step function that results from:

compose the 4th row of the triangle, with alternating signs. This is a generalization of the following basic result (often used in electrical engineering):

is the boxcar function.[24] The corresponding row of the triangle is row 0, which consists of just the number 1.

If n is congruent to 2 or to 3 mod 4, then the signs start with −1. In fact, the sequence of the (normalized) first terms corresponds to the powers of i, which cycle around the intersection of the axes with the unit circle in the complex plane:

Elementary cellular automaton

The pattern produced by an elementary cellular automaton using rule 60 is exactly Pascal's triangle of binomial coefficients reduced modulo 2 (black cells correspond to odd binomial coefficients).[25] Rule 102 also produces this pattern when trailing zeros are omitted. Rule 90 produces the same pattern but with an empty cell separating each entry in the rows.

Extensions

Pascal's triangle can be extended to negative row numbers.

First write the triangle in the following form:

m
n
0 1 2 3 4 5 ...
0 1 0 0 0 0 0 ...
1 1 1 0 0 0 0 ...
2 1 2 1 0 0 0 ...
3 1 3 3 1 0 0 ...
4 1 4 6 4 1 0 ...

Next, extend the column of 1s upwards:

m
n
0 1 2 3 4 5 ...
−4 1 ...
−3 1 ...
−2 1 ...
−1 1 ...
0 1 0 0 0 0 0 ...
1 1 1 0 0 0 0 ...
2 1 2 1 0 0 0 ...
3 1 3 3 1 0 0 ...
4 1 4 6 4 1 0 ...

Now the rule:

can be rearranged to:

which allows calculation of the other entries for negative rows:

m
n
0 1 2 3 4 5 ...
−4 1 −4 10 −20 35 −56 ...
−3 1 −3 6 −10 15 −21 ...
−2 1 −2 3 −4 5 −6 ...
−1 1 −1 1 −1 1 −1 ...
0 1 0 0 0 0 0 ...
1 1 1 0 0 0 0 ...
2 1 2 1 0 0 0 ...
3 1 3 3 1 0 0 ...
4 1 4 6 4 1 0 ...

This extension preserves the property that the values in the mth column viewed as a function of n are fit by an order m polynomial, namely

.

This extension also preserves the property that the values in the nth row correspond to the coefficients of (1 + x)n:

For example:

When viewed as a series, the rows of negative n diverge. However, they are still Abel summable, which summation gives the standard values of 2n. (In fact, the n = -1 row results in Grandi's series which "sums" to 1/2, and the n = -2 row results in another well-known series which has an Abel sum of 1/4.)

Another option for extending Pascal's triangle to negative rows comes from extending the other line of 1s:

m
n
−4 −3 −2 −1 0 1 2 3 4 5 ...
−4 1 0 0 0 0 0 0 0 0 0 ...
−3 1 0 0 0 0 0 0 0 0 ...
−2 1 0 0 0 0 0 0 0 ...
−1 1 0 0 0 0 0 0 ...
0 0 0 0 0 1 0 0 0 0 0 ...
1 0 0 0 0 1 1 0 0 0 0 ...
2 0 0 0 0 1 2 1 0 0 0 ...
3 0 0 0 0 1 3 3 1 0 0 ...
4 0 0 0 0 1 4 6 4 1 0 ...

Applying the same rule as before leads to

m
n
−4 −3 −2 −1 0 1 2 3 4 5 ...
−4 1 0 0 0 0 0 0 0 0 0 ...
−3 −3 1 0 0 0 0 0 0 0 0 ...
−2 3 −2 1 0 0 0 0 0 0 0 ...
−1 −1 1 −1 1 0 0 0 0 0 0 ..
0 0 0 0 0 1 0 0 0 0 0 ...
1 0 0 0 0 1 1 0 0 0 0 ...
2 0 0 0 0 1 2 1 0 0 0 ...
3 0 0 0 0 1 3 3 1 0 0 ...
4 0 0 0 0 1 4 6 4 1 0 ...

Note that this extension also has the properties that just as

we have

Also, just as summing along the lower-left to upper-right diagonals of the Pascal matrix yields the Fibonacci numbers, this second type of extension still sums to the Fibonacci numbers for negative index.

Either of these extensions can be reached if we define

and take certain limits of the gamma function, .

See also

References

  1. ^ Maurice Winternitz, History of Indian Literature, Vol. III
  2. ^ J. L. Coolidge, The Story of the Binomial Theorem, Amer. Math. Monthly, Vol. 56, No. 3 (Mar., 1949), pp. 147–157
  3. ^ Peter Fox (1998). Cambridge University Library: the great collections. Cambridge University Press. p. 13. ISBN 978-0-521-62647-7.
  4. ^ The binomial coefficient is conventionally set to zero if k is either less than zero or greater than n.
  5. ^ Pascal's triangle | World of Mathematics Summary
  6. ^ A. W. F. Edwards. Pascal's arithmetical triangle: the story of a mathematical idea. JHU Press, 2002. Pages 30–31.
  7. ^ a b c d e f g h i Edwards, A. W. F. (2013), "The arithmetical triangle", in Wilson, Robin; Watkins, John J. (eds.), Combinatorics: Ancient and Modern, Oxford University Press, pp. 166–180.
  8. ^ Alexander Zawaira; Gavin Hitchcock (2008). A Primer for Mathematics Competitions. Oxford University Press. p. 237. ISBN 978-0-19-156170-2.
  9. ^ Selin, Helaine (2008-03-12). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science & Business Media. p. 132. Bibcode:2008ehst.book.....S. ISBN 9781402045592.
  10. ^ The developpement of Arabic Mathematics Between Arithmetic and Algebra - R. Rashed "Page 63"
  11. ^ Sidoli, Nathan; Brummelen, Glen Van (2013-10-30). From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren. Springer Science & Business Media. p. 54. ISBN 9783642367366.
  12. ^ Kennedy, E. (1966). Omar Khayyam. The Mathematics Teacher 1958. National Council of Teachers of Mathematics. p. 140–142. JSTOR i27957284.
  13. ^ Coolidge, J. L. (1949), "The story of the binomial theorem", The American Mathematical Monthly, 56 (3): 147–157, doi:10.2307/2305028, JSTOR 2305028, MR 0028222.
  14. ^ Weisstein, Eric W. (2003). CRC concise encyclopedia of mathematics, p. 2169. ISBN 978-1-58488-347-0.
  15. ^ Smith, Karl J. (2010), Nature of Mathematics, Cengage Learning, p. 10, ISBN 9780538737586.
  16. ^ Fowler, David (January 1996). "The Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17. doi:10.2307/2975209. JSTOR 2975209. See in particular p. 11.
  17. ^ Brothers, H. J. (2012), "Finding e in Pascal's triangle", Mathematics Magazine, 85: 51, doi:10.4169/math.mag.85.1.51.
  18. ^ Brothers, H. J. (2012), "Pascal's triangle: The hidden stor-e", The Mathematical Gazette, 96: 145–148, doi:10.1017/S0025557200004204.
  19. ^ Foster, T. (2014), "Nilakantha's Footprints in Pascal's Triangle", Mathematics Teacher, 108: 247, doi:10.5951/mathteacher.108.4.0246
  20. ^ Fine, N. J. (1947), "Binomial coefficients modulo a prime", American Mathematical Monthly, 54 (10): 589–592, doi:10.2307/2304500, JSTOR 2304500, MR 0023257. See in particular Theorem 2, which gives a generalization of this fact for all prime moduli.
  21. ^ Hinz, Andreas M. (1992), "Pascal's triangle and the Tower of Hanoi", The American Mathematical Monthly, 99 (6): 538–544, doi:10.2307/2324061, JSTOR 2324061, MR 1166003. Hinz attributes this observation to an 1891 book by Édouard Lucas, Théorie des nombres (p. 420).
  22. ^ Wolfram, S. (1984). "Computation Theory of Cellular Automata". Comm. Math. Phys. 96 (1): 15–57. Bibcode:1984CMaPh..96...15W. doi:10.1007/BF01217347.
  23. ^ For a similar example, see e.g. Hore, P. J. (1983), "Solvent suppression in Fourier transform nuclear magnetic resonance", Journal of Magnetic Resonance, 55 (2): 283–300, Bibcode:1983JMagR..55..283H, doi:10.1016/0022-2364(83)90240-8.
  24. ^ Karl, John H. (2012), An Introduction to Digital Signal Processing, Elsevier, p. 110, ISBN 9780323139595.
  25. ^ Wolfram, S. (2002). A New Kind of Science. Champaign IL: Wolfram Media. pp. 870, 931–2.

External links

(2,1)-Pascal triangle

In mathematics, the (2,1)-Pascal triangle (mirrored Lucas triangle)is a triangular array.

The rows of the (2,1)-Pascal triangle (sequence A029653 in the OEIS) are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows.

The triangle is based on the Pascal's Triangle with the second line being (2,1) and the first cell of each row set to 2.

This construction is related to the binomial coefficients by Pascal's rule, with one of the terms being .

Binomial coefficient

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers nk ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula

For example, the fourth power of 1 + x is

and the binomial coefficient is the coefficient of the x2 term.

Arranging the numbers in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation

The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol is usually read as "n choose k" because there are ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are ways to choose 2 elements from namely and

The binomial coefficients can be generalized to for any complex number z and integer k ≥ 0, and many of their properties continue to hold in this more general form.

Central binomial coefficient

In mathematics the nth central binomial coefficient is the particular binomial coefficient

They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at n = 0 are:

1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...; (sequence A000984 in the OEIS)
Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables.

For example, in

the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant coefficient. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term.

Often coefficients are numbers as in this example, although they could be parameters of the problem or any expression in these parameters. In such a case one must clearly distinguish between symbols representing variables and symbols representing parameters. Following René Descartes, the variables are often denoted by x, y, ..., and the parameters by a, b, c, ..., but it is not always the case. For example, if y is considered as a parameter in the above expression, the coefficient of x is −3y, and the constant coefficient is 1.5 + y.

When one writes

it is generally supposed that x is the only variable and that a, b and c are parameters; thus the constant coefficient is c in this case.

Similarly, any polynomial in one variable x can be written as

for some positive integer , where are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient. For the largest with (if any), is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial

is 4.

Some specific coefficients that occur frequently in mathematics have received a name. This is the case of the binomial coefficients, the coefficients which occur in the expanded form of , and are tabulated in Pascal's triangle.

Community (season 1)

The first season of the television comedy series Community originally aired from September 17, 2009 on NBC to May 20, 2010 in the United States. The first three episodes aired at 9:30 pm ET before being moved to 8:00 pm ET. The show was picked up for 22 episodes in October 2009, and an additional 3 episodes were ordered later.

The show focuses on disbarred lawyer Jeff Winger, and his attempt to get a bachelor's degree at a community college, while he forms a bond with his Spanish study group.

Constructible polygon

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known.

Gould's sequence

Gould's sequence is an integer sequence named after Henry W. Gould that counts the odd numbers in each row of Pascal's triangle. It consists only of powers of two, and begins:

1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, ... (sequence A001316 in the OEIS)For instance, the sixth number in the sequence is 4, because there are four odd numbers in the sixth row of Pascal's triangle (the four bold numbers in the sequence 1, 5, 10, 10, 5, 1).

Halayudha

Halayudha (Sanskrit: हलायुध) was a 10th-century Indian mathematician who wrote the Mṛtasañjīvanī, a commentary on Pingala's Chandaḥśāstra. The latter contains a clear description of Pascal's triangle (called meru-prastaara).

Hexany

The hexany, invented by Erv Wilson, represents one of the simplest structures found in his Combination Product Sets. It is referred to as an uncentered structure, meaning that it implies no tonic. It achieves this by using consonant relations as opposed to the dissonance methods normally employed by atonality. While it is often and confused overlapped with the Euler Fokker Genus, the subsequent stellation of Wilson's Combination product sets (CPS) are outside of that Genus. The Euler Fokker Genus fails to see 1 as a possible member of a set except as a starting point. The numbers of vertices of his combination sets follow the numbers in Pascal's triangle. In this construction, the hexany is the third cross-section of the four-factor set and the first uncentered one. hexany is the name that Erv Wilson gave to the six notes in the 2-out-of-4 combination product set, abbreviated as 2*4 CPS.The hexany can be thought of as analogous to the octahedron. The notes are arranged so that each point represents a pitch and every edge and interval with each face represents a triad. It thus has eight just intonation triads where each triad has two notes in common with three of the other chords. Each triad occurs just once with its inversion represented by the opposing 3 tones. The edges of the octahedron show musical intervals between the vertices, usually chosen to be consonant intervals from the harmonic series. The points represent musical notes, and the three notes that make each of the triangular faces represent musical triads. Wilson also pointed out and explored the idea of melodic Hexanies.

Simply, the hexany is the 2 out of 4 set. It is constructed by taking any four factors and a set of two at a time, then multiplying them in pairs. For instance, the harmonic factors 1, 3, 5 and 7 are combined in pairs of 1*3, 1*5, 1*7, 3*5, 3*7, 5*7, resulting in 1, 3, 5, 7 Hexanies. The notes are usually octave shifted to place them all within the same octave, which has no effect on interval relations and the consonance of the triads. The possibility of an octave being a solution is not outside of Wilson's conception and is used in cases of placing larger combination product sets upon Generalized Keyboards.

Leibniz harmonic triangle

The Leibniz harmonic triangle is a triangular arrangement of unit fractions in which the outermost diagonals consist of the reciprocals of the row numbers and each inner cell is the cell diagonally above and to the left minus the cell to the left. To put it algebraically, L(r, 1) = 1/r (where r is the number of the row, starting from 1, and c is the column number, never more than r) and L(r, c) = L(r - 1, c - 1) − L(r, c - 1).

List of factorial and binomial topics

This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation).

Abel's binomial theorem

Alternating factorial

Antichain

Beta function

Bhargava factorial

Binomial coefficient

Binomial distribution

Binomial proportion confidence interval

Binomial-QMF (Daubechies wavelet filters)

Binomial series

Binomial theorem

Pascal's triangle

Binomial transform

Binomial type

Carlson's theorem

Catalan number

Fuss–Catalan number

Central binomial coefficient

Combination

Combinatorial number system

De Polignac's formula

Difference operator

Difference polynomials

Digamma function

Egorychev method

Erdős–Ko–Rado theorem

Euler–Mascheroni constant

Faà di Bruno's formula

Factorial

Factorial moment

Factorial number system

Factorial prime

Gamma distribution

Gamma function

Gaussian binomial coefficient

Gould's sequence

Hyperfactorial

Hypergeometric distribution

Hypergeometric function identities

Hypergeometric series

Incomplete beta function

Incomplete gamma function

Kempner function

Lah number

Lanczos approximation

Lozanić's triangle

Macaulay representation of an integer

Mahler's theorem

Multinomial distribution

Multinomial coefficient, Multinomial formula, Multinomial theorem

Multiplicities of entries in Pascal's triangle

Multiset

Multivariate gamma function

Narayana numbers

Negative binomial distribution

Nörlund–Rice integral

Pascal matrix

Pascal's pyramid

Pascal's simplex

Pascal's triangle

Permutation

List of permutation topics

Pochhammer symbol (also falling, lower, rising, upper factorials)

Poisson distribution

Polygamma function

Primorial

Proof of Bertrand's postulate

Sierpinski triangle

Star of David theorem

Stirling number

Stirling transform

Stirling's approximation

Subfactorial

Table of Newtonian series

Taylor series

Trinomial expansion

Vandermonde's identity

Wilson prime

Wilson's theorem

Wolstenholme prime

List of triangle topics

This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrices, or concretely in physical space. It does not include metaphors like "love triangle" in which the word has no reference to the geometric shape.

Lozanić's triangle

Lozanić's triangle (sometimes called Losanitsch's triangle) is a triangular array of binomial coefficients in a manner very similar to that of Pascal's triangle. It is named after the Serbian chemist Sima Lozanić, who researched it in his investigation into the symmetries exhibited by rows of paraffins (archaic term for alkanes).

The first few lines of Lozanić's triangle are

listed in (sequence A034851 in the OEIS).

Like Pascal's triangle, outer edge diagonals of Lozanić's triangle are all 1s, and most of the enclosed numbers are the sum of the two numbers above. But for numbers at odd positions k in even-numbered rows n (starting the numbering for both with 0), after adding the two numbers above, subtract the number at position (k − 1)/2 in row n/2 − 1 of Pascal's triangle.

The diagonals next to the edge diagonals contain the positive integers in order, but with each integer stated twice OEIS: A004526.

Moving inwards, the next pair of diagonals contain the "quarter-squares" (OEIS: A002620), or the square numbers and pronic numbers interleaved.

The next pair of diagonals contain the alkane numbers l(6, n) (OEIS: A005993). And the next pair of diagonals contain the alkane numbers l(7, n) (OEIS: A005994), while the next pair has the alkane numbers l(8, n) (OEIS: A005995), then alkane numbers l(9, n) (OEIS: A018210), then l(10, n) (OEIS: A018211), l(11, n) (OEIS: A018212), l(12, n) (OEIS: A018213), etc.

The sum of the nth row of Lozanić's triangle is (OEIS: A005418 lists the first thirty values or so).

The sums of the diagonals of Lozanić's triangle intermix with (where Fx is the xth Fibonacci number).

As expected, laying Pascal's triangle over Lozanić's triangle and subtracting yields a triangle with the outer diagonals consisting of zeroes (OEIS: A034852, or OEIS: A034877 for a version without the zeroes). This particular difference triangle has applications in the chemical study of catacondensed polygonal systems.

Multinomial distribution

In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for rolling a k-sided die n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

When k is 2 and n is 1, the multinomial distribution is the Bernoulli distribution. When k is 2 and n is bigger than 1, it is the binomial distribution. When k is bigger than 2 and n is 1, it is the categorical distribution.

The Bernoulli distribution models the outcome of a single Bernoulli trial. In other words, it models whether flipping a (possibly biased) coin one time will result in either a success (obtaining a head) or failure (obtaining a tail). The binomial distribution generalizes this to the number of heads from performing n independent flips (Bernoulli trials) of the same coin. The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a categorical distribution, such as rolling a k-sided die n times.

Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p1, ..., pk, and n independent trials. Since the k outcomes are mutually exclusive and one must occur we have pi ≥ 0 for i = 1, ..., k and . Then if the random variables Xi indicate the number of times outcome number i is observed over the n trials, the vector X = (X1, ..., Xk) follows a multinomial distribution with parameters n and p, where p = (p1, ..., pk). While the trials are independent, their outcomes X are dependent because they must be summed to n.

In some fields such as natural language processing, categorical and multinomial distributions are synonymous and it is common to speak of a multinomial distribution when a categorical distribution is actually meant. This stems from the fact that it is sometimes convenient to express the outcome of a categorical distribution as a "1-of-K" vector (a vector with one element containing a 1 and all other elements containing a 0) rather than as an integer in the range ; in this form, a categorical distribution is equivalent to a multinomial distribution over a single trial.

Pascal's pyramid

In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's Pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial numbers and relates to the binomial expansion and the binomial distribution. The binomial and trinomial numbers, coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names.

Pentatope number

A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left.

The first few numbers of this kind are :

1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 (sequence A000332 in the OEIS)

Pentatope numbers belong in the class of figurate numbers, which can be represented as regular, discrete geometric patterns. The formula for the nth pentatopic number is:

Two of every three pentatope numbers are also pentagonal numbers. To be precise, the (3k − 2)th pentatope number is always the ((3k2 − k)/2)th pentagonal number and the (3k − 1)th pentatope number is always the ((3k2 + k)/2)th pentagonal number. The 3kth pentatope number is the generalized pentagonal number obtained by taking the negative index −(3k2 + k)/2 in the formula for pentagonal numbers. (These expressions always give integers).

The infinite sum of the reciprocals of all pentatopal numbers is . This can be derived using telescoping series.

Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.In biochemistry, they represent the number of possible arrangements of n different polypeptide subunits in a tetrameric (tetrahedral) protein. No prime is the predecessor of a pentatope number, and the largest semiprime which is the predecessor of a pentatope number is 1819.

Similarly, the only primes preceding a 6-simplex number are 83 and 461.

Sierpiński triangle

The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or Sierpinski sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets–that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

Singmaster's conjecture

Singmaster's conjecture is a conjecture in combinatorial number theory in mathematics, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than the number 1, which appears infinitely many times). It is clear that the only number that appears infinitely many times in Pascal's triangle is 1, because any other number x can appear only within the first x + 1 rows of the triangle.

Yang Hui

Yang Hui (simplified Chinese: 杨辉; traditional Chinese: 楊輝; pinyin: Yáng Huī, ca. 1238–1298), courtesy name Qianguang (謙光), was a late-Song dynasty Chinese mathematician from Qiantang (modern Hangzhou, Zhejiang). Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the other famous mathematician Qin Jiushao.

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