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In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra.In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, [1] India, [2] China, Germany, and Italy.
A level-5 approximation to a Sierpiński triangle obtained by shading the first 2 5 (32) levels of a Pascal's triangle white if the binomial coefficient is even and black otherwise If one takes Pascal's triangle with 2 n {\displaystyle 2^{n}} rows and colors the even numbers white, and the odd numbers black, the result is an approximation to ...
1000th row of Pascal's triangle, arranged vertically, with grey-scale representations of decimal digits of the coefficients, right-aligned. The left boundary of the image corresponds roughly to the graph of the logarithm of the binomial coefficients, and illustrates that they form a log-concave sequence .
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix , an upper-triangular matrix , or a symmetric matrix .
Having an algorithm to print out some rows of the triangle doesn't have much to do with Pascal's triangle. I originally put in the Python because it replaced a Java program that was about 10 times as long; but in any event having a program is optional, so let's just cut it and avoid the language wars. Wile E. Heresiarch 06:47, 6 Mar 2005 (UTC)
The powers of two that divide the central binomial coefficients are given by Gould's sequence, whose nth element is the number of odd integers in row n of Pascal's triangle. Squaring the generating function gives 1 1 − 4 x = ( ∑ n = 0 ∞ ( 2 n n ) x n ) ( ∑ n = 0 ∞ ( 2 n n ) x n ) . {\displaystyle {\frac {1}{1-4x}}=\left(\sum _{n=0 ...
Pascal's triangle, rows 0 through 7. The hockey stick identity confirms, for example: for n =6, r =2: 1+3+6+10+15=35. In combinatorics , the hockey-stick identity , [ 1 ] Christmas stocking identity , [ 2 ] boomerang identity , Fermat's identity or Chu's Theorem , [ 3 ] states that if n ≥ r ≥ 0 {\displaystyle n\geq r\geq 0} are integers, then
In mathematics, the Bell triangle is a triangle of numbers analogous to Pascal's triangle, whose values count partitions of a set in which a given element is the largest singleton. It is named for its close connection to the Bell numbers , [ 1 ] which may be found on both sides of the triangle, and which are in turn named after Eric Temple Bell .