cp's OEIS Frontend

A161565 E.g.f. satisfies: A(x) = exp(x*exp(2*x*A(x))).

%I A161565 #8 Nov 17 2017 19:23:45
%S A161565 1,1,5,37,417,6201,115393,2583141,67643201,2029868785,68699859201,
%T A161565 2589393498429,107580709769569,4885086832499433,240716442970201409,
%U A161565 12793428673619226901,729511897042502788737,44427614415877495608801
%N A161565 E.g.f. satisfies: A(x) = exp(x*exp(2*x*A(x))).
%H A161565 G. C. Greubel, <a href="/A161565/b161565.txt">Table of n, a(n) for n = 0..367</a>
%F A161565 a(n) = Sum_{k=0..n} 2^(n-k) * C(n,k) * (n-k+1)^(k-1) * k^(n-k).
%F A161565 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 A161565 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! +...
%e A161565 log(A(x)) = x*B(x) where B(x) = exp(2*x*A(x)) = e.g.f. of A161566:
%e A161565 B(x) = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! + 10362*x^5/5! +...
%t A161565 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 A161565 (PARI) {a(n)=sum(k=0,n,2^(n-k)*binomial(n,k) * (n-k+1)^(k-1) * k^(n-k))}
%o A161565 (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 A161565 Cf. A161566.
%K A161565 nonn
%O A161565 0,3
%A A161565 _Paul D. Hanna_, Jun 14 2009