A356950 E.g.f. satisfies log(A(x)) = x^3 * (exp(x) - 1) * A(x).

1, 0, 0, 0, 24, 60, 120, 210, 60816, 544824, 3175920, 14969790, 1339209960, 25141598196, 291418089144, 2618105492730, 128974591028640, 3841451570440560, 73103023032142176, 1060951475511351414, 39132892925113341240, 1516348247446904304300

Offset: 0

Author

Seiichi Manyama, Sep 06 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A052880, A355843, A356949.

Cf. A000272, A356099, A356952.

Programs

Mathematica

nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])*A[x]*x^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*stirling(n-3*k, k, 2)/(n-3*k)!);

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^3*(exp(x)-1))^k/k!)))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3*(1-exp(x))))))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3*(1-exp(x)))/(x^3*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * Stirling2(n-3*k,k)/(n-3*k)!.

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

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

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