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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
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 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 is an imperative and procedural programming language, designed by Niklaus Wirth as a small, efficient language intended to encourage good programming practices using structured programming and data structuring. It is named after French mathematician, philosopher and physicist Blaise Pascal. [a]
This is an accepted version of this page This is the latest accepted revision, reviewed on 17 February 2025. General-purpose programming language "C programming language" redirects here. For the book, see The C Programming Language. Not to be confused with C++ or C#. C Logotype used on the cover of the first edition of The C Programming Language Paradigm Multi-paradigm: imperative (procedural ...
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
Furthermore, the entries of this triangle can be computed from Pascal's: "The terms in each row are the initial term divided by the corresponding Pascal triangle entries." [ 2 ] In fact, each diagonal relates to corresponding Pascal Triangle diagonals: The first Leibniz diagonal consists of 1/(1x natural numbers ), the second of 1/(2x ...
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