A177777 - OEIS

%I #8 Mar 17 2017 16:45:45

%S 1,2,12,152,3640,160224,13063792,2012388736,596666619648,

%T 344964885948160,392058233038486784,880255154481199466496,

%U 3916538634445633156373504,34603083354426212294072477696

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

%C An analog of the LambertW function.

%C A053549 without the leading term. - _R. J. Mathar_, May 24 2010

%H Vaclav Kotesovec, <a href="/A177777/b177777.txt">Table of n, a(n) for n = 1..70</a>

%F a(n) = n*A001187(n), where A001187(n) is the number of connected labeled graphs with n nodes.

%F Let B(x) = Sum_{n>=0} 2^(n(n-1)/2)*x^n/n! then

%F . L(x) = x*d/dx log(B(x)) = x*B'(x)/B(x) and

%F . 1/B(x) = Sum_{n>=0} (-1)^n/n!*Product_{k=0..n-1} L(2^k*x).

%e E.g.f.: L(x) = x + 2*x^2/2! + 12*x^3/3! + 152*x^4/4! + 3640*x^5/5! +...

%e which is invariant under the series:

%e 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 Let B(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 64*x^4/4! + 1024*x^5/5! +...

%e 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 then L(x) = x*d/dx log(B(x)) which also satisfies:

%e 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 (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 nonn

%O 1,2

%A _Paul D. Hanna_, May 19 2010