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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.
Jia Xian triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD. Yang Hui referred to Jia Xian's Shi Suo Suan Shu in the Yongle Encyclopedia Jia Xian ( simplified Chinese : 贾宪 ; traditional Chinese : 賈憲 ; pinyin : Jiǎ Xiàn ; Wade–Giles : Chia Hsien ; ca. 1010–1070) was a Chinese ...
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
Binomial coefficients C (n, k) extended for negative and fractional n, illustrated with a simple binomial. It can be observed that Pascal's triangle is rotated and alternate terms are negated. The case n = −1 gives Grandi's series. For any n,
The following is an APL one-liner function to visually depict Pascal's triangle: Pascal ← { ' ' @ ( 0 =⊢ ) ↑ 0 , ⍨¨ a ⌽ ¨ ⌽∊ ¨ 0 , ¨¨ a ∘ ! ¨ a ← ⌽⍳ ⍵ } ⍝ Create a one-line user function called Pascal Pascal 7 ⍝ Run function Pascal for seven rows and show the results below: 1 1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 6 ...
Pascal's triangle, whose entries are the binomial coefficients [8] Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called generalized Pascal triangles; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers. [9]
I agree with Wile that the code does not add any information on Pascal's triangle. The algorithm, based on Pascal's identity is already explained in English in the lead section, and it is straightforward to translate it in a specific programming language. -- Jitse Niesen 15:23, 6 Mar 2005 (UTC) I've cut the section with the computer code.
Layers of Pascal's pyramid derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial – the number of terms is clearly a triangular number. In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by