faktoriaaln1

Faktoriaaln1 is a generalized factorial concept defined for positive integers. In this article, it denotes the product of all positive integers up to n that satisfy a fixed parity condition, specifically that they are odd. Formally, faktoriaaln1(n) = ∏_{k=1, k≤n, k odd}^n k. For example, faktoriaaln1(7) = 1·3·5·7 = 105.

Relation to double factorial: faktoriaaln1(n) equals the odd double factorial. If n = 2m+1, faktoriaaln1(n) = (2m+1)!!. If n = 2m, faktoriaaln1(n) = (2m-1)!!. There are closed forms in terms of standard factorials: (2m+1)!! = (2m+1)! / (2^m m!).

Generalizations: The concept can be extended by choosing different residue classes modulo a fixed q, yielding families of generalized factorials. Faktoriaaln1 denotes the case q = 2 with residue 1, i.e., selecting all odd integers up to n.

Applications and computation: Faktoriaaln1 grows rapidly and appears in combinatorial counting and in teaching generalized products as an illustrative example. It can be computed by iterative multiplication. Asymptotic approximations can leverage Stirling-type formulas for factorials together with the relation to double factorials to estimate Faktoriaaln1(n) for large n.

See also: Factorial, Double factorial, Generalized factorials, Stirling's approximation, Gamma function.