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%I #9 Nov 18 2017 01:40:51
%S 1,2,16,206,3976,101402,3237220,124293206,5582747824,287346080690,
%T 16680250440124,1078327289938670,76840445565238024,
%U 5984507179839282122,505778795448930860308,46104043794638089809158
%N E.g.f. satisfies: A(x) = exp(2*x*exp(3*x*A(x))).
%H G. C. Greubel, <a href="/A161568/b161568.txt">Table of n, a(n) for n = 0..341</a>
%F a(n) = Sum_{k=0..n} C(n,k) * 2^k * 3^(n-k) * (n-k+1)^(k-1) * k^(n-k).
%F More generally, if G(x) = exp(p*x*exp(q*x*G(x))) = Sum_{n>=0} g(n)*x^n/n!,
%F then g(n) = Sum_{k=0..n} C(n,k) * p^k * q^(n-k) * (n-k+1)^(k-1) * k^(n-k).
%F a(n) ~ sqrt(s/3) * n^(n-1) / (exp(n) * r^(n+1/2)), where r = 0.149417197143691584817... and s = 2.468671804906329807069... are roots of the system of equations 3*r*s*Log(s) = 1, 6*exp(3*r*s)*s*r^2 = 1. - _Vaclav Kotesovec_, Jul 15 2014
%e E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 206*x^3/3! + 3976*x^4/4! +...
%t Flatten[{1,Table[Sum[Binomial[n,k] * 2^k * 3^(n-k) * (n-k+1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jul 15 2014 *)
%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)*2^k*3^(n-k)*(n-k+1)^(k-1)*k^(n-k))}
%o (PARI) {a(n)=local(A=1+x);for(i=0,n,A=exp(2*x*exp(3*x*A+O(x^n))));n!*polcoeff(A,n,x)}
%Y Cf. A161565, A161566, A161567.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 14 2009
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