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%I #17 Sep 12 2022 05:18:33
%S 1,0,1,3,26,230,2794,39564,663606,12712104,275171106,6632699040,
%T 176309074644,5123121177096,161577261004860,5497133655605760,
%U 200683752698028924,7825434930630743616,324616635150708044796,14273994548639305751040,663205761925601097418488
%N E.g.f. satisfies A(x) = (1 - x * A(x))^(log(1 - x * A(x)) / 2).
%F E.g.f. satisfies log(A(x)) = log(1 - x * A(x))^2 / 2.
%F a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (n+1)^(k-1) * |Stirling1(n,2*k)|/(2^k * k!).
%t m = 21; (* number of terms *)
%t A[_] = 0;
%t Do[A[x_] = (1 - x*A[x])^(Log[1 - x*A[x]]/2) + O[x]^m // Normal, {m}];
%t CoefficientList[A[x], x]*Range[0, m - 1]! (* _Jean-François Alcover_, Sep 12 2022 *)
%o (PARI) a(n) = sum(k=0, n\2, (2*k)!*(n+1)^(k-1)*abs(stirling(n, 2*k, 1))/(2^k*k!));
%Y Cf. A001761, A357037.
%Y Cf. A347001, A357028.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Sep 09 2022
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