%I #27 May 21 2022 08:30:14
%S 1,2,20,392,12560,579872,36034880,2874676352,284538241280,
%T 34058188677632,4831480473359360,799233222752602112,
%U 152126941229960990720,32947584100184816033792,8042650107769696199720960,2194728130327915760239542272,664779526701915421094019399680
%N E.g.f. exp(sinh(x)^2) (even powers only).
%H Alois P. Heinz, <a href="/A009236/b009236.txt">Table of n, a(n) for n = 0..264</a>
%F a(n) = Sum_{k=1..n}(Sum_{i=0..2*k} (-1)^i*(k-i)^(2*n)*binomial(2*k,i))*(2^(2*(n-k)) /k!)), n>0, a(0)=1. - _Vladimir Kruchinin_, Jun 06 2011
%F a(n) = 4^n*Sum_{k=0..2*n} (-1)^k*binomial(2*n, k)*B(k, 1/4)*B(2*n-k, 1/4) where B(n, x) are the Bell polynomials. - _Peter Luschny_, Sep 10 2017
%p A009236 := n -> 4^n*(add(binomial(2*n, k)*(-1)^k*BellB(k, 1/4)*BellB(2*n-k, 1/4), k = 0..2*n)): seq(A009236(n), n=0..16); # _Peter Luschny_, Sep 10 2017
%p # second Maple program:
%p b:= proc(n) option remember; `if`(n=0, 1, add(
%p b(n-2*j)*binomial(n-1, 2*j-1)*2^(2*j-1), j=1..n/2))
%p end:
%p a:= n-> b(2*n):
%p seq(a(n), n=0..16); # _Alois P. Heinz_, Jun 21 2021
%t b[n_] := b[n] = If[n == 0, 1, Sum[b[n-2*j]*Binomial[n-1, 2*j-1]*2^(2*j-1), {j, 1, n/2}]];
%t a[n_] := b[2*n];
%t Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, May 21 2022, after _Alois P. Heinz_ *)
%o (Maxima) a(n):=sum(sum((-1)^i*(k-i)^(2*n)*binomial(2*k,i),i,0,2*k)*(2^(2*(n-k))/k!),k,1,n); /* _Vladimir Kruchinin_, Jun 06 2011 */
%o (PARI) my(x='x+O('x^40), v = Vec(serlaplace(exp(sinh(x)^2)))); vector(#v\2, k, v[2*k-1]) \\ _Michel Marcus_, May 21 2022
%K nonn
%O 0,2
%A _R. H. Hardin_
%E Extended and signs tested by _Olivier Gérard_, Mar 15 1997