The Double Factorial Explained: What n!! Means and Where It Shows Up

The first time I wrote two exclamation marks after a number in a homework answer, my teacher circled it and asked whether I meant the factorial of a factorial. I did not. I meant the double factorial, written n!!, and that small confusion is exactly why this operation deserves a clear explanation. It is not (n!)!. It is a single, well-defined product that turns up in places ordinary factorials cannot reach.

What a double factorial actually computes

The double factorial of n multiplies every other integer down from n until you hit the bottom of its parity chain. That single sentence is the whole idea. Instead of stepping down by one at each multiplication the way a normal factorial does, you step down by two.

That leaves you with two cases depending on whether n is odd or even:

If n is odd, the chain runs through the odd numbers and ends at one: n × (n−2) × (n−4) × … × 3 × 1.
If n is even, the chain runs through the even numbers and ends at two: n × (n−2) × (n−4) × … × 4 × 2.

So the concrete rule is simple: n!! multiplies every other integer down from n, keeping only the numbers that share n's parity. There is no mixing. An odd input never picks up an even factor, and an even input never reaches one.

Two boundary values round things out. By convention 0!! = 1 and (−1)!! = 1, both because they are empty products — there is nothing to multiply, and the agreed value of an empty product is one. These are not arbitrary. They keep the recurrence n!! = n × (n−2)!! consistent right down to the floor, the same way 0! = 1 keeps the ordinary factorial recurrence intact.

A worked example: 7!! = 105

Take 7!!. Start at seven and step down by two until you land on one:

7!! = 7 × 5 × 3 × 1 = 105

That is the entire computation. Seven times five is thirty-five, times three is one hundred and five, times one is still one hundred and five. The chain has four factors and never touches an even number.

Compare that to the ordinary factorial of the same input:

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

The factorial uses every integer and balloons to 5040, while the double factorial keeps only the odd ones and lands on 105. Run the same exercise on an even number and the pattern holds the other way: 8!! = 8 × 6 × 4 × 2 = 384, ending at two rather than one. If you want to see the full expansion line for any input, the double factorial calculator writes out every factor and the final product.

How n!! differs from the ordinary factorial

The visual similarity between ! and !! hides how differently they grow. The factorial keeps every term; the double factorial throws away half of them. For the same n, n!! is always far smaller than n!, and the gap widens fast as n climbs.

There is also a tidy bridge between the two, which is worth knowing because it lets you convert one into the other:

For even n = 2k: n!! = 2^k × k!. For example, 8!! = 8 × 6 × 4 × 2 = (2·4)(2·3)(2·2)(2·1) = 2⁴ × 4! = 16 × 24 = 384.
For odd n: multiply the odd and even chains together to recover the full factorial. Since (2k)! = (2k)!! × (2k−1)!!, you get (2k−1)!! = (2k)! ÷ (2^k × k!).

If you ever need the standalone factorial side of that identity, the factorial calculator handles n! directly, and you can multiply or divide the two results yourself to confirm the bridge. The relationship is the reason the double factorial is sometimes called the semifactorial — it is, in a sense, half of the factorial's work.

Where double factorials earn their keep

The double factorial is not a textbook curiosity. It shows up wherever a formula naturally selects every other term, and three areas lean on it constantly.

Combinatorics. The number of ways to partition 2k objects into unordered pairs — perfect matchings — is exactly (2k−1)!!. With six people pairing off for a project, that is 5!! = 5 × 3 × 1 = 15 distinct pairings. The double factorial counts these directly, with no extra division by symmetry factors, because the skipping already removes the overcounting.

Probability. The even moments of a standard normal distribution are odd double factorials. The expected value of X raised to the power 2k equals (2k−1)!!. So the fourth moment is 3!! = 3, the sixth is 5!! = 15, and the eighth is 7!! = 105. That last number is the same 105 from the worked example above, which is a nice reminder that these are not separate facts but the same operation wearing different hats.

Physics integrals. When you integrate even powers of sine or cosine over a quarter period, the Wallis formula expresses the answer as a ratio of double factorials, typically (n−1)!! / n!! with a π/2 factor for even powers. Because that closed form is built from exact integer products, keeping the double factorials exact keeps the final integral a clean fraction instead of a decimal you then have to reverse-engineer.

Common slip-ups to avoid

A few mistakes recur often enough to call out:

Treating !! as factorial applied twice. 7!! is one operation that steps by two and equals 105. (7!)! would be the factorial of 5040, a number with thousands of digits. The two marks mean "skip every other term", not "do it again".
Mixing parity in the chain. An odd input must end at one; an even input must end at two. Writing 8!! = 8 × 6 × 4 × 2 × 1 wrongly drags a one onto an even chain, and 7!! = 7 × 5 × 3 drops the final one. Check the last factor against the parity of n.
Forgetting the boundary values. Both 0!! and (−1)!! equal one, not zero. Treating 0!! as zero quietly breaks the moment identity and the Wallis ratio, because those formulas divide by it.

Once the parity rule clicks, the double factorial is one of the friendlier pieces of notation in mathematics: a single product, stepping down by two, ending exactly where its parity sends it. The next time two exclamation marks show up in a probability or physics problem, you will know it is not asking for a factorial twice over — just for every other term, multiplied down to the floor.

Made by Toolora · Updated 2026-06-13