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%I #14 Jul 15 2022 21:59:37
%S 1,1,4,32,393,6547,138046,3525853,105832964,3651748332,142429413387,
%T 6196895235709,297571887174040,15632879134292045,891910713837242092,
%U 54919409605089141532,3630105859259972654905,256374187841461047791587
%N E.g.f. satisfies: log(A(x)) = (1 - exp(-x*A(x))) * A(x).
%H Alois P. Heinz, <a href="/A349558/b349558.txt">Table of n, a(n) for n = 0..360</a>
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * (n+k+1)^(k-1) * Stirling2(n,k).
%F a(n) ~ sqrt((s-1)*s^3 / (1 + r*(2*s - 3)*s - r^2*(s-1)*s^2)) * n^(n-1) / (exp(n) * r^(n -1/2)), where r = 0.2202409288542107090687589144963703329896230236509... and s = 1.7315644042495989781932730410872588555151921253414... are roots of the system of equations s = s/exp(r*s) + log(s), (s-1)/s - (1 - r*s)/exp(r*s) = 0. - _Vaclav Kotesovec_, Nov 22 2021
%t a[n_] := Sum[(-1)^(n - k) * (n + k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 22 2021 *)
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(n+k+1)^(k-1)*stirling(n, k, 2));
%Y Cf. A058864, A349557.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Nov 21 2021
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