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

1, 2, 12, 152, 3640, 160224, 13063792, 2012388736, 596666619648, 344964885948160, 392058233038486784, 880255154481199466496, 3916538634445633156373504, 34603083354426212294072477696

Offset: 1

Author

Paul D. Hanna, May 19 2010

Keywords

• nonn

Comments

An analog of the LambertW function.

A053549 without the leading term. - R. J. Mathar, May 24 2010

Examples

			E.g.f.: L(x) = x + 2*x^2/2! + 12*x^3/3! + 152*x^4/4! + 3640*x^5/5! +...
which is invariant under the series:
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! +...
Let B(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 64*x^4/4! + 1024*x^5/5! +...
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! +...
then L(x) = x*d/dx log(B(x)) which also satisfies:
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! -+...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..70

Programs

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)}

Formula

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

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

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

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