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A374882
Expansion of e.g.f. exp( (1 - (1 - 9*x)^(1/3))/3 ).
2
1, 1, 7, 109, 2665, 88981, 3768391, 193406977, 11663021329, 808092594505, 63252127883431, 5519514702282901, 531266903931402937, 55912682968563924829, 6387276499619184590695, 787104141893585220839401, 104074098535487279656795681, 14697203663694095986066104337
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Bell Polynomial.
FORMULA
a(n) = Sum_{k=0..n} (-9)^(n-k) * Stirling1(n,k) * A317996(k) = (-9)^n * Sum_{k=0..n} (1/3)^k * Stirling1(n,k) * Bell_k(-1/3), where Bell_n(x) is n-th Bell polynomial.
From Vaclav Kotesovec, Aug 02 2024: (Start)
a(n) = 18*(n-2)*a(n-1) - 9*(3*n-8)*(3*n-7)*a(n-2) + a(n-3).
a(n) ~ Gamma(1/3) * 3^(2*n - 3/2) * n^(n - 5/6) / (sqrt(2*Pi) * exp(n - 1/3)) * (1 - 2*Pi/(3^(3/2)*Gamma(1/3)^2*n^(1/3))). (End)
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1-(1-9*x)^(1/3))/3)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 02 2024
STATUS
approved