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In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. It is named after the French mathematician Blaise Pascal in much of the Western world, although other mathematicians studied it centuries before him in India,^{[1]} Iran, China, Germany, and Italy.^{[2]}
The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top. 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. A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number above and to the left with the number above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first 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.
This construction is related to the binomial coefficients by Pascal's rule, which says that if
then
for any nonnegative integer n and any integer k between 0 and n.^{[3]}
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 set of numbers that form Pascal's triangle were known before Pascal. However, Pascal developed many uses of it and was the first one to organize all the information together in his treatise, Traité du triangle arithmétique (1653). The numbers originally arose from Indian studies of combinatorics and binomial numbers and the Greeks' study of figurate numbers.^{[4]}
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.^{[5]}^{[6]} While Pingala's work only survives in fragments, the commentator Varāhamihira, around 505, gave a clear description of the additive formula,^{[6]} 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.^{[6]} In approximately 850, the Jain mathematician Mahāvīra gave a different formula for the binomial coefficients, using multiplication, equivalent to the modern formula .^{[6]} 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.^{[6]}
At around the same time, it was discussed in Persia (Iran) by the Persian mathematician, AlKaraji (953–1029).^{[7]} It was later repeated by the Persian poetastronomermathematician Omar Khayyám (1048–1131); thus the triangle is referred to as the KhayyamPascal triangle or Khayyam triangle in Iran. 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.^{[8]}
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 in China.^{[9]}
In the west, the binomial coefficients were calculated by Gersonides in the early 14th century, using the multiplicative formula for them.^{[6]} 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.^{[10]} 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.^{[6]} In Italy, Pascal's triangle is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolò Fontana Tartaglia (1500–77), who published six rows of the triangle in 1556.^{[6]} Gerolamo Cardano, also, published the triangle as well as the additive and multiplicative rules for constructing it in 1570.^{[6]}
Pascal's Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published posthumously in 1665. 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.^{[11]}
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} (the figure above and to the left of the figure to be determined, since it is on the same diagonal) + a_{i−1} (the figure to the immediate right of the first figure). 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.
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. For example, suppose a basketball team has 10 players and wants to know how many ways there are of selecting 8. Provided we have the first row and the first entry in a row numbered 0, the answer is entry 8 in row 10: 45. That is, the solution of 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.
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  
1  8  28  56  70  56  28  8  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 (the original triangle possesses none) + 1 (built upon the single face of the original triangle) = 1; the number of faces is 1 (the original triangle itself) + 3 (the new faces, each built upon an edge of the original triangle) = 4; the number of edges is 3 (from the original triangle) + 3 (the new edges, each built upon a vertex of the original triangle) = 6; the number of new vertices is 3 (from the original triangle) + 1 (the new vertex that was added to create the tetrahedron from the triangle) = 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. Alternately, the "extra" 1 in a row can be thought of as the empty space, or field, in which the vertices, edges, faces, etc., exist.
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:
1 1 2 1 4 4 1 6 12 8 1 8 24 32 16 1 10 40 80 80 32 1 12 60 160 240 192 64 1 14 84 280 560 672 448 128
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 − 1}. Again, to use the elements of row 5 as an example: , which is equal to .
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.^{[16]} 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.^{[17]} 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).^{[18]} 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 = 0  m = 1  m = 2  m = 3  m = 4  m = 5  ...  
n = 0  1  0  0  0  0  0  ... 
n = 1  1  1  0  0  0  0  ... 
n = 2  1  2  1  0  0  0  ... 
n = 3  1  3  3  1  0  0  ... 
n = 4  1  4  6  4  1  0  ... 
Next, extend the column of 1s upwards:
m = 0  m = 1  m = 2  m = 3  m = 4  m = 5  ...  
n = −4  1  ...  
n = −3  1  ...  
n = −2  1  ...  
n = −1  1  ...  
n = 0  1  0  0  0  0  0  ... 
n = 1  1  1  0  0  0  0  ... 
n = 2  1  2  1  0  0  0  ... 
n = 3  1  3  3  1  0  0  ... 
n = 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 = 0  m = 1  m = 2  m = 3  m = 4  m = 5  ...  
n = −4  1  −4  10  −20  35  −56  ... 
n = −3  1  −3  6  −10  15  −21  ... 
n = −2  1  −2  3  −4  5  −6  ... 
n = −1  1  −1  1  −1  1  −1  ... 
n = 0  1  0  0  0  0  0  ... 
n = 1  1  1  0  0  0  0  ... 
n = 2  1  2  1  0  0  0  ... 
n = 3  1  3  3  1  0  0  ... 
n = 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 = −4  m = −3  m = −2  m = −1  m = 0  m = 1  m = 2  m = 3  m = 4  m = 5  ...  
n = −4  1  0  0  0  0  0  0  0  0  0  ... 
n = −3  1  0  0  0  0  0  0  0  0  ...  
n = −2  1  0  0  0  0  0  0  0  ...  
n = −1  1  0  0  0  0  0  0  ...  
n = 0  0  0  0  0  1  0  0  0  0  0  ... 
n = 1  0  0  0  0  1  1  0  0  0  0  ... 
n = 2  0  0  0  0  1  2  1  0  0  0  ... 
n = 3  0  0  0  0  1  3  3  1  0  0  ... 
n = 4  0  0  0  0  1  4  6  4  1  0  ... 
Applying the same rule as before leads to
m = −4  m = −3  m = −2  m = −1  m = 0  m = 1  m = 2  m = 3  m = 4  m = 5  ...  
n = −4  1  0  0  0  0  0  0  0  0  0  ... 
n = −3  −3  1  0  0  0  0  0  0  0  0  ... 
n = −2  3  −2  1  0  0  0  0  0  0  0  ... 
n = −1  −1  1  −1  1  0  0  0  0  0  0  .. 
n = 0  0  0  0  0  1  0  0  0  0  0  ... 
n = 1  0  0  0  0  1  1  0  0  0  0  ... 
n = 2  0  0  0  0  1  2  1  0  0  0  ... 
n = 3  0  0  0  0  1  3  3  1  0  0  ... 
n = 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, .
Meeting: 1003, Atlanta, Georgia, SS 24A, AMS Special Session on Design Theory and Graph Theory, I 100306607 Avery S. Zoch, Pascal’s Grey Scale
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