A370940 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - 2*log(1+x)) ).

1, 2, 14, 184, 3612, 94968, 3139088, 125181936, 5851551680, 313874206656, 19006905318528, 1282738818650496, 95477483835672960, 7770589670409684480, 686519279618695022592, 65436589709543394150912, 6693486627002144059422720, 731378220534326743907266560

Offset: 0

Author

Seiichi Manyama, Mar 06 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A258922, A370941.

Programs

PARI

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

PARI

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

Formula

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

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