A356797 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^2.

1, 0, 2, 3, 64, 305, 6936, 64897, 1645008, 24290289, 692240680, 14243244521, 456748635432, 12105737521033, 435619742434800, 14112089558682585, 567134312211275296, 21653262317886286817, 966207399513747354072, 42358800314758614030505

Offset

0,3

Links

Table of n, a(n) for n=0..19.

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.

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

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

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

Mathematica

m = 20; (* number of terms *)
CoefficientList[Exp[-(1/2)*LambertW[-2*(Exp[x]-1)*x]] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 11 2022 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*(1-exp(x)))/2)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*(1-exp(x)))/(2*x*(1-exp(x))))^(1/2)))

Crossrefs

Cf. A052506, A355843, A356798.

Cf. A356788.

Sequence in context: A280582 A037426 A004854 * A356785 A015169 A041953

Adjacent sequences: A356794 A356795 A356796 * A356798 A356799 A356800

Keyword

nonn

Author

Seiichi Manyama, Aug 28 2022

Status

approved