A357244 E.g.f. satisfies A(x) * log(A(x)) = 2 * (exp(x) - 1).

1, 2, -2, 22, -266, 4614, -102442, 2777030, -88914730, 3283693254, -137408080298, 6425417730758, -332055079469610, 18792899306652358, -1156017201432075946, 76796076655220486854, -5479395288838822143786, 417905042599836811225798, -33928512587303405767179178

Offset

0,2

Links

Table of n, a(n) for n=0..18.

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

Formula

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

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

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

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

Mathematica

nmax = 18; A[_] = 1;
Do[A[x_] = Exp[(2*(Exp[x] - 1))/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

Prog

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

Crossrefs

Cf. A349583, A357245.

Cf. A356908.

Sequence in context: A118326 A212847 A087405 * A001012 A040082 A014358

Adjacent sequences: A357241 A357242 A357243 * A357245 A357246 A357247

Keyword

sign

Author

Seiichi Manyama, Sep 19 2022

Status

approved