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

%I A262020 #45 Apr 12 2019 13:30:00
%S A262020 1,0,1,-1,5,-11,43,-127,489,-1693,6771,-26071,109693,-457757,2028671,
%T A262020 -9039931,42101329,-198411489,967906675,-4791497559,24401815141,
%U A262020 -126243354637,669094876055,-3603105436163,19818039219577,-110721426757801,630419303537115
%N A262020 Inverse binomial transform of double factorial n!! = A006882(n).
%H A262020 Alois P. Heinz, <a href="/A262020/b262020.txt">Table of n, a(n) for n = 0..800</a>
%F A262020 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)).
%F A262020 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.
%F A262020 a(n) = Sum_{k=0..n} (-1)^k * C(n,k) *  A006882(n-k).
%F A262020 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
%F A262020 G.f.: Sum_{k>=0} k!!*x^k/(1 + x)^(k+1). - _Ilya Gutkovskiy_, Apr 12 2019
%p A262020 a:= proc(n) option remember; `if`(n<3, (n-1)^2,
%p A262020        (n-2)*a(n-3) +(n-1)*a(n-2) -2*a(n-1))
%p A262020     end:
%p A262020 seq(a(n), n=0..30);
%t A262020 Table[Sum[(-1)^(n-k) * Binomial[n, k] * k!!, {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Oct 31 2017 *)
%Y A262020 Cf. A000166 (the same for n!), A006882, A263529.
%K A262020 sign
%O A262020 0,5
%A A262020 _Alois P. Heinz_, Oct 22 2015