Enter any non-negative integer n into the Double Factorial Calculator and get n!! — the product of all positive integers up to n that share the same parity (odd or even) as n. The result field shows the computed double factorial value, plus a step-by-step breakdown of the multiplication sequence so you can see exactly how the answer is reached.
Double Factorial (n!!)
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Multiplication Sequence
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Parity of n
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Number of Terms Multiplied
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The double factorial of n (written n!!) is the product of all positive integers from n down to 1 (or 2) that have the same parity as n. If n is even, you multiply all even numbers down to 2; if n is odd, you multiply all odd numbers down to 1. For example, 8!! = 8 × 6 × 4 × 2 = 384 and 7!! = 7 × 5 × 3 × 1 = 105.
Start with n and repeatedly subtract 2, multiplying each value together, until you reach either 1 (for odd n) or 2 (for even n). The formula is n!! = n × (n−2) × (n−4) × … continuing until you reach 1 or 2. Both 0!! and 1!! equal 1 by convention.
No — n!! is NOT the same as (n!)!. The notation n!! means the double factorial of n (every other integer multiplied together), whereas (n!)! would mean the factorial of the factorial of n, which grows astronomically larger. For example, 6!! = 48, but (6!)! = 720! which is an enormous number.
By convention, 0!! = 1. This is the empty product, similar to how 0! = 1 in standard factorial notation. The value is defined this way to keep mathematical formulas consistent.
The first double factorial values are: 0!!=1, 1!!=1, 2!!=2, 3!!=3, 4!!=8, 5!!=15, 6!!=48, 7!!=105, 8!!=384, 9!!=945, 10!!=3840. Even and odd sequences grow at different rates.
The notation !!n (with exclamation marks on the left) does NOT represent the double factorial. Double factorial is always written n!! with the symbols after the number. The left-side notation is non-standard and has no widely accepted mathematical meaning.
Double factorials appear in combinatorics, probability theory, and physics. They arise naturally when counting permutations of certain symmetric groups, in the formula for the volume of hyperspheres, in Gaussian integrals, and in quantum mechanics calculations involving spin states.
The double factorial n!! skips every other integer (step of 2). Multifactorials generalize this concept: a triple factorial n!!! skips by 3 (e.g. 9!!! = 9×6×3), a quadruple factorial skips by 4, and so on. The double factorial is the most common and well-studied multifactorial variant.