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A177777
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E.g.f. satisfies: L(x) = x*Sum_{n>=0} (1/n!)*Product_{k=0..n-1} L(2^k*x).
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4
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1, 2, 12, 152, 3640, 160224, 13063792, 2012388736, 596666619648, 344964885948160, 392058233038486784, 880255154481199466496, 3916538634445633156373504, 34603083354426212294072477696
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OFFSET
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1,2
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COMMENTS
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An analog of the LambertW function.
A053549 without the leading term. - R. J. Mathar, May 24 2010
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 1..70
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FORMULA
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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).
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EXAMPLE
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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! -+...
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PROG
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(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)}
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CROSSREFS
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Sequence in context: A208582 A000795 A085628 * A053549 A139383 A216351
Adjacent sequences: A177774 A177775 A177776 * A177778 A177779 A177780
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, May 19 2010
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STATUS
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approved
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