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A353998
Expansion of e.g.f. 1/(1 - x^2/2 * (exp(x) - 1)).
8
1, 0, 0, 3, 6, 10, 195, 1281, 5908, 68076, 758565, 6486535, 75598446, 1059484218, 13378016743, 185273328345, 2999003869800, 48665352612376, 816394913567433, 15110162148144267, 292156921946387170, 5805684093139498470, 122617308231635240331
OFFSET
0,4
FORMULA
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).
a(n) = n! * Sum_{k=0..floor(n/3)} k! * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
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
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2/2*(exp(x)-1))))
(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;
(PARI) a(n) = n!*sum(k=0, n\3, k!*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 13 2022
STATUS
approved