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A356813
Expansion of e.g.f. exp(x * (1 - exp(3*x))).
5
1, 0, -6, -27, 0, 1215, 12312, 45927, -657072, -15857937, -167699160, -266960529, 29356170984, 700068823623, 8419188469104, -1491045413265, -2856006296224992, -79065447339366945, -1162293393139510824, -744123842820101745, 538503788896323210360
OFFSET
0,3
LINKS
FORMULA
G.f.: Sum_{k>=0} (-x)^k / (1 - (3*k+1)*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (3*k+1)^(n-k) * binomial(n,k).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * 3^(n-k) * Stirling2(n-k,k)/(n-k)!.
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-exp(3*x)))))
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(3*k+1)*x)^(k+1)))
(PARI) a(n) = sum(k=0, n, (-1)^k*(3*k+1)^(n-k)*binomial(n, k));
(PARI) a(n) = n!*sum(k=0, n\2, (-1)^k*3^(n-k)*stirling(n-k, k, 2)/(n-k)!);
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Aug 29 2022
STATUS
approved