A377360 E.g.f. satisfies A(x) = ( 1 - log(1 - x*A(x)) )^2.

1, 2, 12, 130, 2082, 44488, 1192964, 38557860, 1459988440, 63414711072, 3108861424032, 169829819311392, 10230860299538400, 673850170929176928, 48176129912775680160, 3715759452364764485280, 307545698210584533055488, 27190399275422185989742080, 2557448587458299889542868480

Offset: 0

Author

Seiichi Manyama, Oct 26 2024

Keywords

• nonn
• changed

Links

• Index entries for reversions of series

Crossrefs

Cf. A138013, A377361.

Cf. A097629, A367080, A376392.

Programs

Mathematica

nmax = 20; CoefficientList[1/x * InverseSeries[Series[x/(1 - Log[1 - x])^2, {x, 0, nmax + 1}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 27 2025 *)

PARI

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367080.

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

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

a(n) ~ sqrt(2) * LambertW(-1, -2*exp(-3))^n * (2 + LambertW(-1, -2*exp(-3)))^(n+2) * n^(n-1) / (exp(n) * sqrt(-1 - LambertW(-1, -2*exp(-3)))). - Vaclav Kotesovec, Aug 27 2025