Multiply integers of the same parity as n while stepping downward by two. The result panel keeps double factorial and its numerical trail together.
Double factorials occur in sphere formulas, repeated derivatives, Gaussian moments, combinatorial matchings, and trigonometric integrals.
The double factorial n!! multiplies n, n−2, n−4, and so on, ending at one for odd n or two for positive even n. Both 0!! and 1!! equal one.
Use Nonnegative integer n as the first Double Factorial checkpoint. Confirm the fixed condition, then anticipate Double factorial. Repeat Double Factorial without reading the prior answer. If the fixed condition differs, preserve both Double Factorial versions and both values of Double factorial.
For n=9, 9!!=9·7·5·3·1=945. For comparison, 9! includes the even factors too and equals 362,880.
Two exclamation marks describe a step of two, not applying factorial twice. Parity determines the terminal factor and must remain visible in hand work.
Increasing n by two multiplies the result by the new n; increasing by one switches parity and creates a different sequence. That behavior gives the double factorial output a built-in reasonableness test.
Ordinary factorial uses every intervening integer and therefore grows faster at comparable inputs. Writing “Double factorial” beside the output prevents that mix-up.
Write the descending sequence with difference two, verify that every factor has the same parity as n, and multiply the factors exactly. Double Factorial also leads to ordinary factorial.
The number from Double Factorial is incomplete without its convention. Record the roles Nonnegative integer n, together with whether positions are distinguishable, events may overlap, or trials are replaced. Those facts define the sample space and cannot be recovered reliably from a copied result alone.
For Double Factorial, use a boundary case such as zero events, one object, certainty, or impossibility when it belongs to the model. A correct boundary result is a useful defense against an unnoticed indexing or probability-scale error.
Compare Double factorial with the quantity named in the Double Factorial question. Re-read Nonnegative integer n, the fixed condition, and the fixed condition before accepting the number. This noun check catches cases where valid arithmetic produces a related value rather than the requested Double factorial.
No.
It is 1.
At 2.
At 1.