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%I #8 Nov 17 2017 19:23:45
%S 1,1,5,37,417,6201,115393,2583141,67643201,2029868785,68699859201,
%T 2589393498429,107580709769569,4885086832499433,240716442970201409,
%U 12793428673619226901,729511897042502788737,44427614415877495608801
%N E.g.f. satisfies: A(x) = exp(x*exp(2*x*A(x))).
%H G. C. Greubel, <a href="/A161565/b161565.txt">Table of n, a(n) for n = 0..367</a>
%F a(n) = Sum_{k=0..n} 2^(n-k) * C(n,k) * (n-k+1)^(k-1) * k^(n-k).
%F a(n) ~ sqrt(LambertW(1/r)) * n^(n-1) / (2*exp(n)*r^(n+1)), where r = 0.256263163133653382... is the root of the equation 1/LambertW(1/r) = -log(2*r^2) - LambertW(1/r). - _Vaclav Kotesovec_, Feb 28 2014
%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! +...
%e log(A(x)) = x*B(x) where B(x) = exp(2*x*A(x)) = e.g.f. of A161566:
%e B(x) = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! + 10362*x^5/5! +...
%t Flatten[{1,Table[Sum[2^(n-k) * Binomial[n,k] * (n-k+1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 28 2014 *)
%o (PARI) {a(n)=sum(k=0,n,2^(n-k)*binomial(n,k) * (n-k+1)^(k-1) * k^(n-k))}
%o (PARI) {a(n)=local(A=1+x);for(i=0,n,A=exp(x*exp(2*x*A+O(x^n))));n!*polcoeff(A,n,x)}
%Y Cf. A161566.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 14 2009
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