A367137 -id:A367137 - cp's OEIS frontend

A367135 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^3)).

1, 1, 9, 166, 4719, 182326, 8927301, 529922002, 36988772211, 2969132797966, 269488306924833, 27291375956851546, 3050923148547318039, 373187615576953777510, 49580088565083198922845, 7109665420655116517351458, 1094492388113416460752513851

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A000670, A052894, A367134.

Cf. A367137, A367139.

Table[1/(3*n+1)! * Sum[(3*n+k)! * StirlingS2[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

a(n) = sum(k=0, n, (3*n+k)!*stirling(n, k, 2))/(3*n+1)!;

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * Stirling2(n,k).

a(n) ~ 3^(4*n) * LambertW(2*exp(1/3)/3)^(3*n + 1) * n^(n-1) / (sqrt(1 + LambertW(2*exp(1/3)/3)) * 2^(3*n + 1) * exp(n) * (3*LambertW(2*exp(1/3)/3) - 1)^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367139 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^3)).

Original entry on oeis.org

1, 1, 9, 167, 4780, 186004, 9173780, 548563140, 38573633016, 3119384230176, 285237426927552, 29102185296785160, 3277703460197645232, 403931173342682581296, 54066960915411480743520, 7811249803193620134996864, 1211525560869437165319590400

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A007840, A052802, A367138.

Cf. A367135, A367137.

Table[1/(3*n+1)! * Sum[(3*n+k)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

a(n) = sum(k=0, n, (3*n+k)!*abs(stirling(n, k, 1)))/(3*n+1)!;

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * |Stirling1(n,k)|.

a(n) ~ LambertW(3*exp(4))^n * n^(n-1) / (sqrt(3*(1 + LambertW(3*exp(4)))) * exp(n) * (-3 + LambertW(3*exp(4)))^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367136 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x)^2)).

Original entry on oeis.org

1, 1, 5, 50, 764, 15804, 413426, 13094864, 487323000, 20844584760, 1007739144312, 54343954158240, 3234285062655984, 210581685526690464, 14889759832273000320, 1136236597054802033664, 93074880409847175490560, 8146156595011083708521472

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A006252, A198860, A367137.

Cf. A367134, A367138.

Table[1/(2*n+1)! * Sum[(2*n+k)! * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

a(n) = sum(k=0, n, (2*n+k)!*stirling(n, k, 1))/(2*n+1)!;

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} (2*n+k)! * Stirling1(n,k).

a(n) ~ LambertW(2*exp(1))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(1)))) * exp(n) * (2 - LambertW(2*exp(1)))^(3*n + 1)). - Vaclav Kotesovec, Nov 07 2023

cp's OEIS Frontend

A367135 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^3)).

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A367139 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^3)).

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Author

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Mathematica

PARI

Formula

A367136 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x)^2)).

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula