A375176 -id:A375176 - 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)

A375174 Expansion of e.g.f. exp( (1/(1 - 9*x)^(1/3) - 1)/3 ).

Original entry on oeis.org

1, 1, 13, 289, 9073, 367681, 18249661, 1071805393, 72684954049, 5588943933313, 480445784729101, 45656401249018561, 4752397230972673393, 537724197016879848769, 65711109523289467682173, 8624825762253351871394161, 1210085772867351648907603201

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A374882, A375176.

Cf. A004212.

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

a(n) = Sum_{k=0..n} 9^(n-k) * |Stirling1(n,k)| * A004212(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) = 36*(n-2)*a(n-1) - 18*(27*n^2 - 135*n + 172)*a(n-2) + (2916*n^3 - 26244*n^2 + 79056*n - 79703)*a(n-3) - 729*(n-4)*(n-3)*(3*n - 11)*(3*n - 10)*a(n-4).
a(n) ~ 3^(2*n - 1/4) * n^(n - 3/8) / (2*exp(n - 4*n^(1/4)/3^(3/2) + 1/3)) * (1 - 35/(32*sqrt(3)*n^(1/4))). (End)

cp's OEIS Frontend

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

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