A357333 E.g.f. satisfies A(x) = -log(1 - x) * exp(2 * A(x)).

0, 1, 5, 50, 778, 16604, 451668, 14947568, 582982160, 26187136128, 1331445995520, 75589772147328, 4739901861071232, 325353447339098112, 24264683011603485696, 1953776475810372817920, 168924939633683095452672, 15609228287753846217412608

Offset: 0

Author

Seiichi Manyama, Sep 24 2022

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.
• Index entries for reversions of series

Crossrefs

Cf. A052807, A357334.

Cf. A264407, A357321.

Programs

Mathematica

nmax = 17; 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)*abs(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 - n*exp(-1)/2 - 1/2)). - Vaclav Kotesovec, Oct 04 2022

E.g.f.: Series_Reversion( 1 - exp(-x * exp(-2*x)) ). - Seiichi Manyama, Sep 09 2024