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A351737
Expansion of e.g.f. exp( x * (exp(3 * x) - 1) ).
11
1, 0, 6, 27, 216, 2025, 21708, 260253, 3460320, 50395041, 795324420, 13495904829, 244747554912, 4718754452529, 96285948702804, 2071265238290565, 46815054101658432, 1108489016781839169, 27424412680091114628, 707277138662880504045, 18974871706141125008640
OFFSET
0,3
LINKS
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * Stirling2(n-k,k)/(n-k)!.
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (3*k-1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} x^k / (1 - (3*k-1)*x)^(k+1). (End)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(3*x)-1))))
(PARI) a(n) = n!*sum(k=0, n\2, 3^(n-k)*stirling(n-k, k, 2)/(n-k)!);
(PARI) a(n) = sum(k=0, n, (3*k-1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(3*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 20 2022
STATUS
approved