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Pascal's triangle has many properties and contains many patterns of numbers. 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 1 and dark pixels 0. The numbers of compositions of n +1 into k +1 ordered partitions form Pascal's triangle.
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 ...
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 elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
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The two sets of three numbers which the Star of David theorem says have equal greatest common divisors also have equal products. [1] For example, again observing that the element 84 is surrounded by, in sequence, the elements 28, 56, 126, 210, 120, 36, and again taking alternating values, we have 28×126×120 = 2 6 ×3 3 ×5×7 2 = 56×210×36.
The trinomial triangle is a variation of Pascal's triangle. The difference between the two is that an entry in the trinomial triangle is the sum of the three (rather than the two in Pascal's triangle) entries above it:
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