A375174 -id:A375174 - cp's OEIS frontend

A374882 Expansion of e.g.f. exp( (1 - (1 - 9*x)^(1/3))/3 ).

1, 1, 7, 109, 2665, 88981, 3768391, 193406977, 11663021329, 808092594505, 63252127883431, 5519514702282901, 531266903931402937, 55912682968563924829, 6387276499619184590695, 787104141893585220839401, 104074098535487279656795681, 14697203663694095986066104337

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A080893, A252284, A317996.

Cf. A375174, A375176.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1-(1-9*x)^(1/3))/3)))

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)

A375176 Expansion of e.g.f. exp( (exp( (exp(9*x) - 1)/3 ) - 1)/3 ).

Original entry on oeis.org

1, 1, 13, 208, 4132, 99328, 2799073, 90310006, 3281661436, 132615087517, 5897867191525, 286140731152972, 15031839986716483, 849637058684740030, 51389339196926149645, 3310400979718767433801, 226189040323182011660827, 16333609964679285918346633

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A374882, A375174.

Cf. A004212.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((exp((exp(9*x)-1)/3)-1)/3)))

a(n) = Sum_{k=0..n} 9^(n-k) * Stirling2(n,k) * A004212(k) = 9^n * Sum_{k=0..n} (1/3)^k * Stirling2(n,k) * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.

cp's OEIS Frontend

A374882 Expansion of e.g.f. exp( (1 - (1 - 9*x)^(1/3))/3 ).

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