login
Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + log(1-x))^2 ).
4

%I #13 Sep 22 2024 11:15:30

%S 1,2,16,238,5270,156048,5803980,260301564,13679476864,824735208864,

%T 56125075306656,4256136846770400,355933078611032880,

%U 32544591173495688480,3230049230183020829184,345849932418702558032736,39738632934736396340588160,4877326190739889592547393792

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

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

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

%F E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367138.

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

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

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

%Y Cf. A052801, A367138.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 22 2024