A356099 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3 * A(x)).

1, 0, 0, 0, 24, 60, 240, 1260, 68544, 604800, 5508000, 54885600, 1877420160, 32069157120, 499522645440, 7832035411200, 236207887534080, 5868136834560000, 133085307920947200, 2941187195765145600, 91568561750088652800, 2857211689810118860800

Offset: 0

Author

Seiichi Manyama, Sep 03 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A353229, A356911.

Programs

Mathematica

nmax = 21; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^3*A[x]) + 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)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);

PARI

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

PARI

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

PARI

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

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.

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

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

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