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Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th centuries. Stirling's approximation provides an accurate approximation to the factorial of large numbers, showing that it grows more quickly than exponential growth.
The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS). In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0! stands for
However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied.
On the other hand, () is "the number of ways to arrange flags on flagpoles", [8] where all flags must be used and each flagpole can have any number of flags. Equivalently, this is the number of ways to partition a set of size n {\displaystyle n} (the flags) into x {\displaystyle x} distinguishable parts (the poles), with a linear order on the ...
The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation!! = ()!! to give !! = (+)!! +.
In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were discovered by Ivo Lah in 1954. [ 1 ] [ 2 ] Explicitly, the unsigned Lah numbers L ( n , k ) {\displaystyle L(n,k)} are given by the formula involving the binomial coefficient
In mathematics, and more specifically number theory, the hyperfactorial of a positive integer is the product of the numbers of the form from to . Definition [ edit ]
In mathematics, and more particularly in number theory, primorial, denoted by "p n #", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.