A262020 Inverse binomial transform of double factorial n!! = A006882(n).

1, 0, 1, -1, 5, -11, 43, -127, 489, -1693, 6771, -26071, 109693, -457757, 2028671, -9039931, 42101329, -198411489, 967906675, -4791497559, 24401815141, -126243354637, 669094876055, -3603105436163, 19818039219577, -110721426757801, 630419303537115

Offset: 0

Author

Alois P. Heinz, Oct 22 2015

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A000166 (the same for n!), A006882, A263529.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, (n-1)^2,
       (n-2)*a(n-3) +(n-1)*a(n-2) -2*a(n-1))
    end:
seq(a(n), n=0..30);

Mathematica

Table[Sum[(-1)^(n-k) * Binomial[n, k] * k!!, {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 31 2017 *)

Formula

E.g.f.: (2*exp(x^2/2)*x+2+sqrt(2*Pi)*exp(x^2/2)*erf(x/sqrt(2))*x) / (2*exp(x)).

a(n) = (n-2)*a(n-3)+(n-1)*a(n-2)-2*a(n-1) for n>2, a(n) = (n-1)^2 otherwise.

a(n) = Sum_{k=0..n} (-1)^k * C(n,k) * A006882(n-k).

a(n) ~ (-1)^n * (sqrt(Pi) - sqrt(2)) * exp(sqrt(n) - n/2 - 1/4) * n^((n+1)/2) / 2. - Vaclav Kotesovec, Oct 31 2017

G.f.: Sum_{k>=0} k!!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019