Built-in Mathematica Symbol

Factorial2 (!!)

n!!
gives the double factorial of n.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

.

n!! is a product of even numbers for n even, and odd numbers for n odd.

Factorial2 can be evaluated to arbitrary numerical precision.

Factorial2 automatically threads over lists.

EXAMPLES

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Basic Examples (2)

Integer values:

Evaluate numerically:

Integer values:

Evaluate numerically:

In[1]:=

Out[1]=

Scope (5)

Evaluate for large arguments:

Evaluate at negative integers:

Plot double factorial:

Evaluate for complex arguments:

Factorial2 threads element-wise over lists:

Generalizations & Extensions (3)

Infinite arguments give symbolic results:

Series expansion at poles:

Series expansion at infinity (generalized Stirling approximation):

Applications (1)

Plot of the absolute value of Factorial2 in the complex plane:

Properties & Relations (5)

Use FunctionExpand to expand double factorial into Gamma function:

Use FullSimplify to simplify expressions involving double factorials:

Sums involving Factorial2:

Generating function:

Recover the original power series:

Products involving double factorial:

Possible Issues (3)

Large arguments can give results too large to be computed explicitly, even approximately:

Smaller values work:

Machine-number inputs can give high-precision results:

To compute repeated factorial use

instead of

:

Neat Examples (3)

Plot Factorial2 at infinity:

Find the numbers of digits 0 through 9 in 10000!!:

Plot the ratio of doubled factorials over double factorial: