Double Factorial Calculator

About this calculator

The double factorial n!! is the product of integers from n down to 1 or 2, skipping every other one so only numbers of the same parity as n are multiplied. For odd n, n!! = n·(n−2)·(n−4)···3·1 (e.g. 7!! = 7·5·3·1 = 105); for even n, n!! = n·(n−2)···4·2 (e.g. 8!! = 8·6·4·2 = 384). It's a distinct operation from applying factorial twice, (n!)! — double factorial is its own well-defined sequence.

Double factorials appear naturally in combinatorics — for instance, (2n−1)!! counts the number of ways to pair up 2n objects (perfect matchings), which is why it shows up in graph theory and statistical mechanics. They also appear in closed-form solutions to integrals involving powers of sine and cosine (the Wallis formulas), in the Gaussian integral's moment expansions, and in the volume formulas for n-dimensional spheres. Students working through combinatorics, probability or calculus problems use this calculator to compute n!! quickly for any n.