A371370 E.g.f. satisfies A(x) = -log(1 - x/(1 - A(x))^2).

0, 1, 5, 62, 1246, 34734, 1239708, 53958456, 2771832656, 164151829440, 11010949643640, 825134834757936, 68321156113803360, 6194283782068848816, 610322188305019432032, 64936303681095948453120, 7419917758371561069774336, 906217650382400588573066880

Offset: 0

Author

Seiichi Manyama, Mar 20 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A368033, A371371.

Cf. A052842.

Programs

Mathematica

Table[Sum[(2*n+k-2)!/(2*n-1)! * Abs[StirlingS1[n,k]], {k,1,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 10 2024 *)

PARI

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

PARI

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

Formula

E.g.f.: Series_Reversion( (1 - x)^2 * (1 - exp(-x)) ).

a(n) = Sum_{k=1..n} (2*n+k-2)!/(2*n-1)! * |Stirling1(n,k)|.

a(n) ~ LambertW(2*exp(3))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(3)))) * (LambertW(2*exp(3)) - 2)^(3*n-1) * exp(n)). - Vaclav Kotesovec, Sep 10 2024