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Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^2*log(1-x))^2 ).
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%I #12 Sep 23 2024 09:28:48

%S 1,0,0,12,24,80,11160,87696,715680,62337600,1065980160,15842534400,

%T 1109943362880,31591940440320,731706348941568,46767587926752000,

%U 1889337264901632000,61735665488234250240,3896148715287564902400,201584132714100384460800,8661099107269708639948800,567405718655558932535500800

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

%F E.g.f.: B(x)^2, where B(x) is the e.g.f. of A371235.

%F a(n) = (2 * n!/(2*n+2)!) * Sum_{k=0..floor(n/3)} (2*n+k+1)! * |Stirling1(n-2*k,k)|/(n-2*k)!.

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

%o (PARI) a(n) = 2*n!*sum(k=0, n\3, (2*n+k+1)!*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/(2*n+2)!;

%Y Cf. A376385, A376392.

%Y Cf. A371235, A375639.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Sep 22 2024