A356902 E.g.f. satisfies A(x) * log(A(x)) = x * (exp(x) - 1).

1, 0, 2, 3, -8, -55, 276, 4417, -13488, -639567, -248300, 141842921, 797525400, -43103642855, -584650622724, 16366430341185, 436555007091616, -6909610676492959, -368240758971238620, 2371795171252419385, 354876368637537736680, 1050192150132691993161

Offset: 0

Author

Seiichi Manyama, Sep 03 2022

Keywords

• sign

Links

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

Crossrefs

Cf. A355843, A356797, A356798, A356904.

Cf. A349583.

Programs

Mathematica

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

PARI

a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*(exp(x)-1)))))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*(1-exp(x))/lambertw(-x*(1-exp(x)))))

Formula

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

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

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

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