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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.
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 ...
Erv Wilson in his paper The Scales of Mt. Meru [6] observed certain diagonals in Pascal's triangle (see diagram) and drew them on paper in 1993. The Padovan numbers were discovered in 1994. Paul Barry (2004) observed that these diagonals generate the Padovan sequence by summing the diagonal numbers. [7]
The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.
In combinatorial mathematics, block walking is a method useful in thinking about sums of combinations graphically as "walks" on Pascal's triangle.As the name suggests, block walking problems involve counting the number of ways an individual can walk from one corner A of a city block to another corner B of another city block given restrictions on the number of blocks the person may walk, the ...
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
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 multiple charms of Pascal's triangle: 1967 Jan: Dr. Matrix delivers a talk on acrostics: 1967 Feb: Mathematical strategies for two-person contests 1967 Mar: An array of problems that can be solved with elementary mathematical techniques 1967 Apr: The amazing feats of professional mental calculators, and some tricks of the trade 1967 May