Contents
How the double factorial calculator works
The calculator multiplies n, n − 2, n − 4 and so on, stopping at 2 for an even input or 1 for an odd input. It keeps the answer as an exact integer.
Two exclamation marks mean a step of two
Enter a whole number from −1 to 5,000. For a small positive n, open the factor list to see which integers were used. The spaces in a long answer are only for reading; copying returns one uninterrupted number.
The endpoint depends on parity. An even chain ends in 2, while an odd chain ends in 1. The base values are .
Do not read n!! as “take a factorial twice”. For example, 5!! = 5 × 3 × 1 = 15, whereas (5!)! means 120!.
Checks for both parity branches
7!! = 105. The factors are 7 × 5 × 3 × 1. Every even integer is skipped.
8!! = 384. The chain is 8 × 6 × 4 × 2. This is the even branch of the same rule.
25!! = 7,905,853,580,625. Thirteen odd factors already produce a 13-digit result, which is why exact integer arithmetic matters.
Useful identities and the calculator's boundary
Ordinary factorial splits into an even chain and an odd chain, so for n ≥ 0:
For 8, this gives 8! = 8!! × 7!! = 384 × 105 = 40,320. Even arguments also satisfy , while odd arguments satisfy .
This tool uses the elementary finite-product definition. It accepts −1 and non-negative integers, but not fractions or negative odd integers below −1. At the upper limit, 5000!! contains 8,164 digits.
Questions about double factorials
The notation is compact, so most mistakes come from reading it as an ordinary factorial or losing the parity rule.
What is the difference between n! and n!!?
n! multiplies every positive integer down to 1. The double factorial n!! multiplies only integers with the same parity as n, decreasing by 2.
Why is 0!! equal to 1?
It is the base value that makes the even recurrence n!! = n × (n − 2)!! work at n = 2.
Why is (−1)!! equal to 1?
It closes the odd recurrence in the same way, so 1!! = 1 × (−1)!! = 1.
Can I enter a fraction?
No. This calculator covers the finite product for integers from −1 through 5,000.
Are long results rounded?
No. The calculation uses exact integer arithmetic, and the copy button preserves every digit without grouping spaces.
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