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A159596 G.f.: A(x) = exp( Sum_{n>=1} [ D^n x/(1-x)^2 ]^n/n ), where differential operator D = x*d/dx.

%I A159596 #7 Nov 17 2023 09:31:20
%S A159596 1,1,5,22,121,863,8476,118131,2361313,67467236,2731757961,
%T A159596 156417295405,12605225573076,1432381581679361,229016092616239411,
%U A159596 51628631138952017332,16402709158903948390585,7351149638643155728435357
%N A159596 G.f.: A(x) = exp( Sum_{n>=1} [ D^n x/(1-x)^2 ]^n/n ), where differential operator D = x*d/dx.
%C A159596 Conjecture: limit_{n->oo} a(n)^(1/n^2) = 2^(1/4). - _Vaclav Kotesovec_, Nov 17 2023
%H A159596 Vaclav Kotesovec, <a href="/A159596/b159596.txt">Table of n, a(n) for n = 0..113</a>
%F A159596 G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k>=1} k^(n+1)*x^k]^n/n ) where A(x) = Sum_{k>=1} a(k)*x^k.
%e A159596 G.f.: A(x) = 1 + x + 5*x^2 + 22*x^3 + 121*x^4 + 863*x^5 +...
%e A159596 log(A(x)) = Sum_{n>=1} [x + 2^(n+1)*x^2 + 3^(n+1)*x^3 +...]^n/n.
%e A159596 D^n x/(1-x)^2 = x + 2^(n+1)*x^2 + 3^(n+1)*x^3 + 4^(n+1)*x^4 +...
%o A159596 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,sum(k=1,n,k^(m+1)*x^k+x*O(x^n))^m/m)));polcoeff(A,n)}
%Y A159596 Cf. A156170, A159597, A159598.
%K A159596 nonn
%O A159596 0,3
%A A159596 _Paul D. Hanna_, May 05 2009