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A349504 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).

%I A349504 #22 Nov 21 2021 02:48:18
%S A349504 1,1,4,36,484,8840,203868,5691308,186612592,7031373264,299397454080,
%T A349504 14218443479328,745142534904480,42717896158340832,2659373970144454080,
%U A349504 178666030775042040000,12884568940594969258752,992750028716940749121792
%N A349504 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).
%H A349504 Seiichi Manyama, <a href="/A349504/b349504.txt">Table of n, a(n) for n = 0..356</a>
%F A349504 a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling1(n,k).
%F A349504 E.g.f. A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (log(1+x))^k / k!.
%F A349504 From _Vaclav Kotesovec_, Nov 20 2021: (Start)
%F A349504 E.g.f.: sqrt(-LambertW(-2*log(1 + x)) / (2*log(1 + x))).
%F A349504 a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n + exp(-1)/4 - 1)). (End)
%t A349504 nmax = 20; CoefficientList[Series[Sqrt[-LambertW[-2*Log[1 + x]]/(2*Log[1 + x])], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 20 2021 *)
%o A349504 (PARI) a(n) = sum(k=0,n, (2*k+1)^(k-1)*stirling(n, k, 1));
%o A349504 (PARI) N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*log(1+x)^k/k!)))
%Y A349504 Cf. A008275, A033917, A264407, A349505, A349524.
%K A349504 nonn
%O A349504 0,3
%A A349504 _Seiichi Manyama_, Nov 20 2021