A357337 E.g.f. satisfies A(x) = log(1 + x) * exp(2 * A(x)).

0, 1, 3, 26, 334, 5964, 135228, 3729872, 121172560, 4532603904, 191869653120, 9067948437888, 473297792213376, 27039987154142208, 1678363256057198592, 112467293224249912320, 8092242817059530284032, 622253112192770288799744

Offset: 0

Author

Seiichi Manyama, Sep 24 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..356
• Eric Weisstein's World of Mathematics, Lambert W-Function.
• Index entries for reversions of series

Crossrefs

Cf. A277489, A357338.

Cf. A349504.

Programs

Mathematica

nmax = 20; A[_] = 0;
Do[A[x_] = Log[1 + x]*Exp[2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

PARI

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*log(1+x))/2)))

PARI

a(n) = sum(k=1, n, (2*k)^(k-1)*stirling(n, k, 1));

Formula

E.g.f.: -LambertW(-2 * log(1 + x))/2.

a(n) = Sum_{k=1..n} (2 * k)^(k-1) * Stirling1(n,k).

a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n + exp(-1)/4 - 1/2)). - Vaclav Kotesovec, Oct 04 2022

E.g.f.: Series_Reversion( exp(x * exp(-2*x)) - 1 ). - Seiichi Manyama, Sep 10 2024