A357245 E.g.f. satisfies A(x) * log(A(x)) = 3 * (exp(x) - 1).

1, 3, -6, 84, -1599, 42906, -1477716, 62171661, -3090518556, 177237143040, -11518529575857, 836601742598628, -67156626492464064, 5904119985344031639, -564188922815428792914, 58225175660113940932032, -6453955474121138652732903, 764716767229825444834522086

Offset

0,2

Links

Table of n, a(n) for n=0..17.

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

Formula

a(n) = Sum_{k=0..n} 3^k * (-k+1)^(k-1) * Stirling2(n,k).

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

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

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

Mathematica

nmax = 17; A[_] = 1;
Do[A[x_] = Exp[(3*(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, 3^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)*(3*(exp(x)-1))^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*(exp(x)-1)))))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(3*(exp(x)-1)/lambertw(3*(exp(x)-1))))

Crossrefs

Cf. A349583, A357244.

Sequence in context: A364798 A349875 A331403 * A157197 A363410 A211896

Adjacent sequences: A357242 A357243 A357244 * A357246 A357247 A357248

Keyword

sign

Author

Seiichi Manyama, Sep 19 2022

Status

approved