A007911 a(n) = (n-1)!! - (n-2)!!.

1, 1, 5, 7, 33, 57, 279, 561, 2895, 6555, 35685, 89055, 509985, 1381905, 8294895, 24137505, 151335135, 468934515, 3061162125, 10033419375, 68000295825, 234484536825, 1645756410375, 5943863027025, 43105900812975, 162446292283275, 1214871076343925, 4761954230608575

Offset: 3

Author

J. H. Conway

Keywords

• nonn
• easy

Comments

For n >= 0 let A(n) be the product of the positive integers <= n that have the same parity as n minus the product of the positive integers <= n that have the opposite parity as n. Then a(n) = A(n-1) (for n >= 3). [Peter Luschny, Jul 06 2011]

References

• S. P. Hurd and J. S. McCranie, Quantum factorials. Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994). Congr. Numer. 104 (1994), 19-24.

Links

• T. D. Noe, Table of n, a(n) for n = 3..100

Crossrefs

Cf. A007912.

Programs

Magma

DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((n-1))-DoubleFactorial(n-2): n in [3..30]]; // Vincenzo Librandi, Aug 08 2017

Maple

DDF := proc(n) local R, P, k; R := {$1..n}; P := select(k->k mod 2 = n mod 2, R); mul(k, k = P) - mul(k, k = R minus P) end: A007911 := n -> DDF(n-1); # Peter Luschny, Jul 06 2011
f:= gfun:-rectoproc({(-n+1)*a(2+n)+a(1+n)+n^2*a(n), a(2)=0,a(3)=1}, a(n), remember):
map(f, [$3..100]); # Robert Israel, Aug 08 2017

Mathematica

Table[(n - 1)!! - (n - 2)!!, {n, 3, 30}] (* Vincenzo Librandi, Aug 08 2017 *)

Formula

(n-1)*a(n+2) = a(n+1) + n^2*a(n). - Robert Israel, Aug 08 2017