A356910 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2).

1, 0, 0, 6, 12, 40, -180, -1512, -11760, 142560, 2701440, 37033920, -47472480, -7299227520, -181704466944, -904179830400, 40024286265600, 1774386897454080, 24426730612869120, -217650777809310720, -26326923875473536000, -662608157128469637120

Offset

0,4

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/3)} (-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^2 * log(1-x))^k / k!.

E.g.f.: A(x) = exp( LambertW(-x^2 * log(1-x)) ).

E.g.f.: A(x) = -x^2 * log(1-x)/LambertW(-x^2 * log(1-x)).

Mathematica

nmax = 21; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^2))^(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\3, (-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^2*log(1-x))^k/k!)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^2*log(1-x)))))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^2*log(1-x)/lambertw(-x^2*log(1-x))))

Crossrefs

Cf. A356905, A356911.

Cf. A353228, A355287.

Sequence in context: A307181 A052747 A007121 * A353228 A366752 A351503

Adjacent sequences: A356907 A356908 A356909 * A356911 A356912 A356913

Keyword

sign

Author

Seiichi Manyama, Sep 03 2022

Status

approved