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It separates out the sorted and reversed order of a set of items from the other ("mixed") orders, evaluating the number of mixed orders by subtracting two from the usual product formula for the factorial. The product rule for permutations was also described by 6th-century CE Jain monk Jinabhadra. [2]
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n {\displaystyle n} .
The falling factorial occurs in a formula which represents polynomials using the forward difference operator = (+) , which in form is an exact analogue to Taylor's theorem: Compare the series expansion from umbral calculus
2.4 Modified-factorial denominators. ... See Faulhaber's formula. ... The following is a useful property to calculate low-integer-order polylogarithms recursively in ...
Meserve (1948) [9] states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of the Wallis product.
A two-sum formula can be obtained using one of the symmetric formulae for Stirling numbers in conjunction with the explicit formula for Stirling numbers of the second kind. [ n k ] = ∑ j = n 2 n − k ( j − 1 k − 1 ) ( 2 n − k j ) ∑ m = 0 j − n ( − 1 ) m + n − k m j − k m !
Here, n! denotes the factorial of n. The function f (n) (a) denotes the n th derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and (x − a) 0 and 0! are both defined to be 1. This series can be written by using sigma notation, as in the right side formula. [1]
The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, from a set of n objects. The denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. This formula can also be stated in a recursive form.