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A354000
Expansion of e.g.f. exp(x^2/2 * (exp(x) - 1)).
5
1, 0, 0, 3, 6, 10, 105, 651, 2968, 18936, 152505, 1164295, 9109056, 80012868, 756041377, 7387199925, 75535791360, 816560002576, 9254683835073, 109135702334619, 1338613513677280, 17079079303721820, 226148006163689841, 3100114305453613393, 43935964285680790368
OFFSET
0,4
LINKS
FORMULA
a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(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-1)!/2*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v;
(PARI) a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 13 2022
STATUS
approved