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.
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 nonnegative 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 threedimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices.
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 Meruprastaara, 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 AlKaraji (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 poetastronomermathematician 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]}
Pascal's triangle determines the coefficients which arise in binomial expansions. For an example, consider the expansion
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:
where the coefficients a_{i} 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 y^{n} in these binomial expansions, while the next diagonal corresponds to the coefficient of xy^{n−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
The two summations can be reorganized as follows:
(because of how raising a polynomial to a power works, a_{0} = a_{n} = 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 a_{i}s), 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 upperleft to the lowerright correspond to the same power of x, and that the aterms 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 x^{i} term in the polynomial (x + 1)^{n+1} is equal to a_{i−1} + a_{i}. This is indeed the simple rule for constructing Pascal's triangle rowbyrow.
It is not difficult to turn this argument into a proof (by mathematical induction) of the binomial theorem. Since (a + b)^{n} = b^{n}(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} = 2^{n}, 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 nelement 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.
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.
When divided by 2^{n}, 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.
Pascal's triangle has many properties and contains many patterns of numbers.
The diagonals of Pascal's triangle contain the figurate numbers of simplices:
The symmetry of the triangle implies that the n^{th} ddimensional number is equal to the d^{th} ndimensional number.
An alternative formula that does not involve recursion is as follows:
The geometric meaning of a function P_{d} is: P_{d}(1) = 1 for all d. Construct a ddimensional triangle (a 3dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to P_{d}(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find P_{d}(x), have a total of x dots composing the target shape. P_{d}(x) then equals the total number of dots in the shape. A 0dimensional triangle is a point and a 1dimensional triangle is simply a line, and therefore P_{0}(x) = 1 and P_{1}(x) = x, which is the sequence of natural numbers. The number of dots in each layer corresponds to P_{d − 1}(x).
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.
10  
10  20 
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 
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.
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).
Let's begin by considering the 3rd line of Pascal's triangle, with values 1, 3, 3, 1. A 2dimensional triangle has one 2dimensional element (itself), three 1dimensional elements (lines, or edges), and three 0dimensional 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 3dimensional element (itself), four 2dimensional elements (faces), six 1dimensional elements (edges), and four 0dimensional 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 highdimensioned hypertetrahedrons (known as simplices).
To understand why this pattern exists, one must first understand that the process of building an nsimplex 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.
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 2^{k}, 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 2^{Position Number} = 6 × 2^{2} = 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 2dimensional elements in a 2dimensional cube (a square) is one, the number of 1dimensional elements (sides, or lines) is 4, and the number of 0dimensional 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 ncube from an (n − 1)cube is done by simply duplicating the original figure and displacing it some distance (for a regular ncube, 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 ncube, 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 ncube 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 ncube.
In this triangle, the sum of the elements of row m is equal to 3^{m}. Again, to use the elements of row 4 as an example: 1 + 8 + 24 + 32 + 16 = 81, which is equal to .
Each row of Pascal's triangle gives the number of vertices at each distance from a fixed vertex in an ndimensional cube. For example, in three dimensions, the third row (1 3 3 1) corresponds to the usual threedimensional 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 largernumbered rows correspond to hypercubes in each dimension.
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:
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.
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 2^{n}. (In fact, the n = 1 row results in Grandi's series which "sums" to 1/2, and the n = 2 row results in another wellknown 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 lowerleft to upperright 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, .
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 .
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 n ≥ k ≥ 0 and is written It is the coefficient of the x^{k} 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 x^{2} 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 evennumbered rows in Pascal's triangle. The first few central binomial coefficients starting at n = 0 are:
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 polygonIn 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 sequenceGould'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).
HalayudhaHalayudha (Sanskrit: हलायुध) was a 10thcentury Indian mathematician who wrote the Mṛtasañjīvanī, a commentary on Pingala's Chandaḥśāstra. The latter contains a clear description of Pascal's triangle (called meruprastaara).
HexanyThe 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 crosssection of the fourfactor set and the first uncentered one. hexany is the name that Erv Wilson gave to the six notes in the 2outof4 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 triangleThe 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 topicsThis 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
BinomialQMF (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 topicsThis 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 triangleLozanić'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 evennumbered 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 "quartersquares" (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 F_{x} 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 distributionIn probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for rolling a ksided 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 ksided die n times.
Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p_{1}, ..., p_{k}, and n independent trials. Since the k outcomes are mutually exclusive and one must occur we have p_{i} ≥ 0 for i = 1, ..., k and . Then if the random variables X_{i} indicate the number of times outcome number i is observed over the n trials, the vector X = (X_{1}, ..., X_{k}) follows a multinomial distribution with parameters n and p, where p = (p_{1}, ..., p_{k}). 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 "1ofK" 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 pyramidIn mathematics, Pascal's pyramid is a threedimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's Pyramid is the threedimensional analog of the twodimensional 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 numberA pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5term row 1 4 6 4 1 either from left to right or from right to left.
The first few numbers of this kind are :
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 ((3k^{2} − k)/2)th pentagonal number and the (3k − 1)th pentatope number is always the ((3k^{2} + k)/2)th pentagonal number. The 3kth pentatope number is the generalized pentagonal number obtained by taking the negative index −(3k^{2} + 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 6simplex number are 83 and 461.
Sierpiński triangleThe 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 selfsimilar 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 conjectureSingmaster'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 HuiYang Hui (simplified Chinese: 杨辉; traditional Chinese: 楊輝; pinyin: Yáng Huī, ca. 1238–1298), courtesy name Qianguang (謙光), was a lateSong 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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