A357010 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^3 * A(x).

1, 0, 0, 6, 36, 150, 1620, 24486, 293076, 3843510, 68254740, 1311687366, 25479935316, 552545882070, 13437670215060, 345157499363046, 9370414233900756, 274413997443811830, 8572526271218671380, 281754864204797848326, 9767868351458229261396

Offset

0,4

Links

Table of n, a(n) for n=0..20.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

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

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (exp(x) - 1)^(3*k) / k!.

E.g.f.: A(x) = exp( -LambertW(-(exp(x) - 1)^3) ).

E.g.f.: A(x) = -LambertW(-(exp(x) - 1)^3)/(exp(x) - 1)^3.

a(n) ~ sqrt(1 + exp(1/3)) * 3^n * n^(n-1) / (exp(n-1) * (3*log(1 + exp(1/3)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Sep 27 2023

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(Exp[x] - 1)^3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (3*k)!*(k+1)^(k-1)*stirling(n, 3*k, 2)/k!);
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(exp(x)-1)^(3*k)/k!)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-(exp(x)-1)^3))))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-(exp(x)-1)^3)/(exp(x)-1)^3))

Crossrefs

Cf. A052880, A357009.

Cf. A353664, A357025.

Sequence in context: A001117 A353664 A353774 * A357087 A357025 A357085

Adjacent sequences: A357007 A357008 A357009 * A357011 A357012 A357013

Keyword

nonn

Author

Seiichi Manyama, Sep 09 2022

Status

approved