A356911 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3).

1, 0, 0, 0, 24, 60, 240, 1260, -12096, -120960, -1144800, -11642400, 190270080, 4670265600, 81378198720, 1348668921600, -880532674560, -406217626214400, -13255586359142400, -343166884178227200, -3137937973466572800, 72862796986940620800

Offset

0,5

Links

Table of n, a(n) for n=0..21.

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

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) = -x^3 * log(1-x)/LambertW(-x^3 * log(1-x)).

Mathematica

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

Prog

(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(-x^3*log(1-x)/lambertw(-x^3*log(1-x))))

Crossrefs

Cf. A356905, A356910.

Cf. A353229, A356099.

Sequence in context: A185445 A321840 A052759 * A353229 A366777 A351504

Adjacent sequences: A356908 A356909 A356910 * A356912 A356913 A356914

Keyword

sign

Author

Seiichi Manyama, Sep 03 2022

Status

approved