A371370 - OEIS

%I #12 Sep 10 2024 06:17:26

%S 0,1,5,62,1246,34734,1239708,53958456,2771832656,164151829440,

%T 11010949643640,825134834757936,68321156113803360,6194283782068848816,

%U 610322188305019432032,64936303681095948453120,7419917758371561069774336,906217650382400588573066880

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

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

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

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

%F 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

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

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

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

%Y Cf. A368033, A371371.

%Y Cf. A052842.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 20 2024