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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. 4

%I #7 Nov 17 2023 09:31:20

%S 1,1,5,22,121,863,8476,118131,2361313,67467236,2731757961,

%T 156417295405,12605225573076,1432381581679361,229016092616239411,

%U 51628631138952017332,16402709158903948390585,7351149638643155728435357

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

%C Conjecture: limit_{n->oo} a(n)^(1/n^2) = 2^(1/4). - _Vaclav Kotesovec_, Nov 17 2023

%H Vaclav Kotesovec, <a href="/A159596/b159596.txt">Table of n, a(n) for n = 0..113</a>

%F 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 G.f.: A(x) = 1 + x + 5*x^2 + 22*x^3 + 121*x^4 + 863*x^5 +...

%e log(A(x)) = Sum_{n>=1} [x + 2^(n+1)*x^2 + 3^(n+1)*x^3 +...]^n/n.

%e D^n x/(1-x)^2 = x + 2^(n+1)*x^2 + 3^(n+1)*x^3 + 4^(n+1)*x^4 +...

%o (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 Cf. A156170, A159597, A159598.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 05 2009

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Last modified August 11 17:00 EDT 2024. Contains 375073 sequences. (Running on oeis4.)