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%I #15 May 13 2022 12:21:21
%S 1,0,0,3,6,10,195,1281,5908,68076,758565,6486535,75598446,1059484218,
%T 13378016743,185273328345,2999003869800,48665352612376,
%U 816394913567433,15110162148144267,292156921946387170,5805684093139498470,122617308231635240331
%N Expansion of e.g.f. 1/(1 - x^2/2 * (exp(x) - 1)).
%F a(0) = 1; a(n) = n!/2 * Sum_{k=3..n} 1/(k-2)! * a(n-k)/(n-k)! = binomial(n,2) * Sum_{k=3..n} binomial(n-2,k-2) * a(n-k).
%F a(n) = n! * Sum_{k=0..floor(n/3)} k! * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
%F a(n) ~ 2 * n! / ((4 + 2*r + r^3) * r^n), where r = 1.043121496712693605897520269472163423276582653660720448... is the root of the equation (exp(r)-1)*r^2 = 2. - _Vaclav Kotesovec_, May 13 2022
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2/2*(exp(x)-1))))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!/2*sum(j=3, i, 1/(j-2)!*v[i-j+1]/(i-j)!)); v;
%o (PARI) a(n) = n!*sum(k=0, n\3, k!*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));
%Y Cf. A052848, A353999.
%Y Cf. A351505, A354000.
%K nonn
%O 0,4
%A _Seiichi Manyama_, May 13 2022
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