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A177777 E.g.f. satisfies: L(x) = x*Sum_{n>=0} (1/n!)*Product_{k=0..n-1} L(2^k*x).

%I A177777 #8 Mar 17 2017 16:45:45
%S A177777 1,2,12,152,3640,160224,13063792,2012388736,596666619648,
%T A177777 344964885948160,392058233038486784,880255154481199466496,
%U A177777 3916538634445633156373504,34603083354426212294072477696
%N A177777 E.g.f. satisfies: L(x) = x*Sum_{n>=0} (1/n!)*Product_{k=0..n-1} L(2^k*x).
%C A177777 An analog of the LambertW function.
%C A177777 A053549 without the leading term. - _R. J. Mathar_, May 24 2010
%H A177777 Vaclav Kotesovec, <a href="/A177777/b177777.txt">Table of n, a(n) for n = 1..70</a>
%F A177777 a(n) = n*A001187(n), where A001187(n) is the number of connected labeled graphs with n nodes.
%F A177777 Let B(x) = Sum_{n>=0} 2^(n(n-1)/2)*x^n/n! then
%F A177777 . L(x) = x*d/dx log(B(x)) = x*B'(x)/B(x) and
%F A177777 . 1/B(x) = Sum_{n>=0} (-1)^n/n!*Product_{k=0..n-1} L(2^k*x).
%e A177777 E.g.f.: L(x) = x + 2*x^2/2! + 12*x^3/3! + 152*x^4/4! + 3640*x^5/5! +...
%e A177777 which is invariant under the series:
%e A177777 L(x)/x = 1 + L(x) + L(x)L(2x)/2! + L(x)L(2x)L(4x)/3! + L(x)L(2x)L(4x)L(8x)/4! +...
%e A177777 Let B(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 64*x^4/4! + 1024*x^5/5! +...
%e A177777 so that log(B(x)) = x + x^2/2! + 4*x^3/3! + 38*x^4/4! + 728*x^5/5! +...+ A001187(n)*x^n/n! +...
%e A177777 then L(x) = x*d/dx log(B(x)) which also satisfies:
%e A177777 1/B(x) = 1 - L(x) + L(x)L(2x)/2! - L(x)L(2x)L(4x)/3! + L(x)L(2x)L(4x)L(8x)/4! -+...
%o A177777 (PARI) {a(n,r=1)=local(A=x+x^2);for(i=1,n,A=x*sum(m=0,n,r^m/m!*prod(k=0,m-1,subst(A,x,2^k*x+x*O(x^n)))));n!*polcoeff(A,n)}
%K A177777 nonn
%O A177777 1,2
%A A177777 _Paul D. Hanna_, May 19 2010