login
E.g.f. satisfies A(x) = exp(x/(1 - A(x))^2) - 1.
4

%I #13 Mar 29 2024 05:28:23

%S 0,1,5,61,1209,33261,1171933,50363293,2554659761,149399423101,

%T 9896519640981,732401926901613,59890184672573929,5362586032967290765,

%U 521831581416561627149,54834132144912233219581,6188110724712474697469025,746431260858514472012500701

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

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

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

%F a(n) = Sum_{k=1..n} (2*n+k-2)!/(2*n-1)! * Stirling2(n,k).

%F a(n) ~ 2^(n-1) * LambertW(exp(1/2))^(2*n-1) * n^(n-1) / (sqrt(LambertW(exp(1/2)) + 1) * exp(n) * (2*LambertW(exp(1/2))-1)^(3*n-1)). - _Vaclav Kotesovec_, Mar 29 2024

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

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

%Y Cf. A368033, A371370.

%Y Cf. A052892.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 20 2024