Essays/Double Factorial

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Even Factorial

Even factorial is a product of the first n even positive integers, numbers of the sequence {2,4,6,8,...}. The even factorial of 0 is defined to be 1.

 F_e(n) = \prod_{i=1}^n 2 n, \quad n>0, \quad F_e(0) \equiv 1

By definition, for n=0..10 the values are

   (,: ([:*/2*1+i.)"0) i.11x
0 1 2  3   4    5     6      7        8         9         10
1 2 8 48 384 3840 46080 645120 10321920 185794560 3715891200

We will use an alternative analytical definition F_e(n)=2^nn!, as we need to double each member of the natural sequence {2*1,2*2,2*3,2*4,...} to obtain the even sequence.

   Fe=: 2&^ * !

   (Fe -: ([:*/2*1+i.)"0) i.100x
1

Note: as seen in the example by definition, the assignment of value at 0 conforms with the identity value of product insert for empty arrays.

Odd Factorial

Odd factorial is a product of the first n odd positive integers, numbers of the sequence {1,3,5,7,...}. The odd factorial of 0 is defined to be 1.

 F_o(n) = \prod_{i=1}^n 2 n - 1 = \prod_{i=0}^{n-1} 2 n + 1, \quad n>0, \quad F_o(0) \equiv 1

By definition, for n=0..10 the values are

   (,: ([:*/1+2*i.)"0) i.11x
0 1 2  3   4   5     6      7       8        9        10
1 1 3 15 105 945 10395 135135 2027025 34459425 654729075

We will use an alternative analytical definition F_o(n)=\frac{(2n)!}{2^nn!}=\frac{(2n)!}{F_e(n)}, as we need to reduce the product of the natural sequence twice as long by the product of the even members.

   Fo=: !@+: % 2&^ * !        NB. !@+: % Fe

   (Fo -: ([:*/1+2*i.)"0) i.100x
1

Double Factorial

Double factorial is an alternating sequence of even and odd factorials: if the argument is even, the result comes from the even sequence, if it is odd, from the odd.


n!! \equiv  \left\{
\begin{array}{ll}
 n (n-2) \dots 5 \cdot 3 \cdot 1 & n>0 \quad odd \\
 n (n-2) \dots 6 \cdot 4 \cdot 2 & n>0 \quad even \\
 1, & n=-1,0 \\
\end{array}
    \right.

By definition, we construct the branches for even and odd arguments separately, and for n=-1..14 the values are

   (>:@i.&.-:) 6
2 4 6
   (>:@(i.&.-:)@>:) 5
1 3 5
   (>:@i.&.-:)`(>:@(i.&.-:)@>:)@.(2&|)"0] 5 6
1 3 5
2 4 6
   (,: ([: */ (>:@i.&.-:)`(>:@(i.&.-:)@>:)@.(2&|))"0) <:i.16
_1 0 1 2 3 4  5  6   7   8   9   10    11    12     13     14
 1 1 1 2 3 8 15 48 105 384 945 3840 10395 46080 135135 645120

We will use our alternative analytical definitions for odd and even factorials


n!! \equiv  \left\{
\begin{array}{lll}
 F_o(\frac{n+1}{2}), & n \ge -1 \quad odd \\
 F_e(\frac{n}{2}),   & n \ge 0 \quad even \\
\end{array}
    \right.


Fd=: (Fe@-:)`(Fo@-:@>:)@.(2&|)"0       NB. n!!

Double factorials as alternating odd and even sequences.

   ,./|:_2]\"1  (,: Fd) <:i.16
_1      1  0      1
 1      1  2      2
 3      3  4      8
 5     15  6     48
 7    105  8    384
 9    945 10   3840
11  10395 12  46080
13 135135 14 645120

Properties

   (Fe -: Fd@+:) i.10

1

F_e(n) = (2n)!!
   (Fo -: Fd@<:@+:) i.10

1

F_o(n) = (2n-1)!!
   (Fo -: Fd@>:@+:@<:) i.10

1

F_o(n) = (2(n-1)+1)!!

See Also

  • Bifactorial
  • Generalized Monty Hall
  • OEIS OEIS:A122774 Triangle of bifactorial numbers, n B m = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, read by rows
  • OEIS OEIS:A000165 Double factorial numbers: (2n)!! = 2^n*n! (even)
  • OEIS OEIS:A001147 Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1) (odd)
  • OEIS OEIS:A006882 Double factorials n!!: a(n)=n*a(n-2)
  • MathWorld:DoubleFactorial, Mathworld