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    Multifactorial


    Factorials are not limited to the single factorial (n!) and may be extended to (n!!), (n!!!), (n!!!!), etc.   The double factorial is the most commonly used variant where:

         n!! denotes the double factorial of n and is defined recursively by:

    n!! = 1,   if n = −1 or n = 0 or n = 1

                                                    
    n    !! = n(n − 2)!!,     if n  ≥ 2


    Examples:

    5!! = 5 x 3 x 1 = 15

    6!! = 6 x 4 x 2 = 48

    7!! = 7 x 5 x 3 x 1 = 105

    10!! = 10 x 8 x 6 x 4 x 2 = 3840
     
     Some identities involving double factorials include:



    In general, the kth factorial, denoted by n!(k), is defined recursively as:
                                    
    n!(k) = 1,    if  0 ≤ n < k
                                                    
    n!(k) = n(n - k)!(k),   if n  ≥ k