A356904 E.g.f. satisfies A(x)^2 * log(A(x)) = x * (exp(x) - 1).

1, 0, 2, 3, -32, -175, 2376, 29617, -371440, -9251919, 91421560, 4529155961, -26677647864, -3160004989271, 1541460644192, 2946529440977865, 19556193589426336, -3498019439220155551, -56274505323609293208, 5077223330715030358009

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, A356902.

Cf. A349654.

Programs

Mathematica

nmax = 19; A[_] = 1;
Do[A[x_] = Exp[((Exp[x]-1)*x)/A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

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*(exp(x)-1))/2)))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*(1-exp(x))/lambertw(-2*x*(1-exp(x))))^(1/2)))

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 * (exp(x) - 1))/2 ).

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