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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]
Comparison of Stirling's approximation with the factorial. 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 . It is named after James Stirling, though a related but less precise result was first stated ...
In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n. [1] That is, Restated, this says that for even n, the double factorial [2] is while for odd n it is For example, 9‼ = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ ...
A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences. Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations: where the coefficients are the same as those in the binomial theorem.
Definition. The factorial number system is a mixed radix numeral system: the i -th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value is to be multiplied by (i − 1)! (its place value). Radix/Base. 8.
In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position. In other words, a derangement is a permutation that has no fixed points. The number of derangements of a set of size n is known as the subfactorial of n or the n- th derangement number or n- th de Montmort ...
Stirling numbers of the first kind are the coefficients in the expansion of the falling factorial. into powers of the variable : For example, , leading to the values , , and . Subsequently, it was discovered that the absolute values of these numbers are equal to the number of permutations of certain kinds. These absolute values, which are known ...
Legendre's formula. In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac.