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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.
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,
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.
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, 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
Pascal constant and type defines are built in and don't need a preprocessor. There were programmers using a preprocessor also with Pascal (sometimes the same one used with C), certainly not as common as with C. Although often pointed out as a "lack" in Pascal, technically C does not have program modularity nor macros built in either.
Having an algorithm to print out some rows of the triangle doesn't have much to do with Pascal's triangle. I originally put in the Python because it replaced a Java program that was about 10 times as long; but in any event having a program is optional, so let's just cut it and avoid the language wars. Wile E. Heresiarch 06:47, 6 Mar 2005 (UTC)
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix , an upper-triangular matrix , or a symmetric matrix .