A370936 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - log(1+2*x)/2) ).

1, 1, 2, 8, 48, 384, 3872, 47088, 671360, 10985088, 202927872, 4178030592, 94874787840, 2355758714880, 63498696376320, 1846607063998464, 57630620308930560, 1921296165774950400, 68145277700464312320, 2562234152415762972672, 101801592691389968154624

Offset: 0

Author

Seiichi Manyama, Mar 06 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A198860, A370937.

Programs

Mathematica

a[n_]:=Sum[2^(n-k)*(n+k)!*StirlingS1[n, k],{k,0,n}]/(n+1)!; Array[a,21,0] (* Stefano Spezia, Apr 20 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-log(1+2*x)/2))/x))

PARI

a(n) = sum(k=0, n, 2^(n-k)*(n+k)!*stirling(n, k, 1))/(n+1)!;

Formula

a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^(n-k) * (n+k)! * Stirling1(n,k).

a(n) ~ 2^(2*n + 1) * LambertW(exp(-1))^n * n^(n-1) / (sqrt(1 + LambertW(exp(-1))) * exp(n) * (1 - LambertW(exp(-1)))^(2*n + 1)). - Vaclav Kotesovec, Mar 06 2024