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A161567 E.g.f. satisfies: A(x) = exp(x*exp(x*A(x)^2)).

%I A161567 #8 Nov 17 2017 20:01:56
%S A161567 1,1,3,22,233,3356,61057,1343686,34731377,1031493880,34617603041,
%T A161567 1295705404874,53516386593001,2417918198462404,118628419305036929,
%U A161567 6280926119941402486,356960234149564116833,21674784895404653181680
%N A161567 E.g.f. satisfies: A(x) = exp(x*exp(x*A(x)^2)).
%H A161567 G. C. Greubel, <a href="/A161567/b161567.txt">Table of n, a(n) for n = 0..365</a>
%F A161567 a(n) = Sum_{k=0..n} C(n,k) * (2*(n-k) + 1)^(k-1) * k^(n-k).
%F A161567 E.g.f.: A(x) = F(x)^(1/2) where F(x) = e.g.f. of A161566.
%F A161567 E.g.f.: A(x) = exp(x*G(x)) where G(x) = e.g.f. of A161565.
%F A161567 a(n) ~ n^(n-1) / (2*exp(n)*r^(n+1/2)), 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 A161567 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 233*x^4/4! +...
%e A161567 log(A(x)) = x*G(x) where G(x) = exp(x*A(x)^2) = e.g.f. of A161565:
%e A161567 G(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! + 6201*x^5/5! +...
%e A161567 A(x)^2 = e.g.f. of A161566:
%e A161567 A(x)^2 = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! + 10362*x^5/5! +...
%t A161567 Flatten[{1,Table[Sum[Binomial[n,k] * (2*(n-k) + 1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 28 2014 *)
%o A161567 (PARI) {a(n)=sum(k=0,n,binomial(n,k)*(2*(n-k)+1)^(k-1)*k^(n-k))}
%o A161567 (PARI) {a(n)=local(A=1+x);for(i=0,n,A=exp(x*exp(x*A^2+O(x^n))));n!*polcoeff(A,n,x)}
%Y A161567 Cf. A161565, A161566.
%K A161567 nonn
%O A161567 0,3
%A A161567 _Paul D. Hanna_, Jun 14 2009