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You are free: to share – to copy, distribute and transmit the work; to remix – to adapt the work; Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made.
You are free: to share – to copy, distribute and transmit the work; to remix – to adapt the work; Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made.
The rows of Pascal's triangle are conventionally enumerated starting with row = at the top (the 0th row). The entries in each row are numbered from the left beginning with = 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 ...
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 ...
The first five layers of Pascal's 3-simplex (Pascal's pyramid). Each face (orange grid) is Pascal's 2-simplex (Pascal's triangle). Arrows show derivation of two example terms. In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.
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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 .
Pascal's triangle, rows 0 through 7. The number of odd integers in row i is the i-th number in Gould's sequence. The self-similar sawtooth shape of Gould's sequence. Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of powers of two, and ...