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The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast, [18] in the first work on Faà di Bruno's formula, [19] but referring to a more general concept of products of arithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial. [20]
For m = 1, the formula is ... which converges when Re(x) > 0. Stirling's formula may also be given in convergent form ... for the factorial ...
In mathematics, the falling factorial (sometimes called the descending factorial, [1] ... formula for the ratio of two ... Attribution-ShareAlike 4.0 ...
See Faulhaber's formula. ... Modified-factorial denominators ... Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; ...
The simple formula for the factorial, x! = 1 × 2 × ⋯ × x is only valid when x is a positive integer, and no elementary function has this property, but a good solution is the gamma function () = (+). [1]
That is, the falling factorial, ... In particular, one formula is the inverse of the other, thus: ... 0 1 3 7 15 −3 0 0 1 6 25
The zero double factorial 0‼ = 1 as an empty product. [3] [4] The sequence of double factorials for even n = 0, 2, 4, 6, 8, ... The generalized formula ...
As there is zero X n+1 or X −1 in (1 + X) n, one might extend the definition beyond the above boundaries to include () = when either k > n or k < 0. This recursive formula then allows the construction of Pascal's triangle, surrounded by white spaces where the zeros, or the trivial coefficients, would be.