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A061541
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Number of connected labeled graphs with n nodes and n+2 edges.
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4
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0, 0, 0, 1, 120, 6165, 258125, 10230360, 405918324, 16530124800, 699126562530, 30884683104000, 1428626760992860, 69248819808744576, 3516693960681822375, 186964957159176734720, 10395215954531344335000, 603712553730550509035520, 36575888366817680447745924
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OFFSET
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1,5
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LINKS
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FORMULA
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E.g.f.: W2(x) = (1/48)*T(x)^4*(2 + 28*T(x) - 23*T(x)^2 + 9*T(x)^3 - T(x)^4)/(1 - T(x))^6, where T(x) is the e.g.f. for rooted labeled trees (A000169), i.e., T(x) = -LambertW(-x) = x*exp(T(x)).
a(n) ~ 5*n^(n+5/2)*sqrt(2*Pi)/256 * (1 - 56*sqrt(2)/(9*sqrt(Pi*n))). - Vaclav Kotesovec, Apr 06 2014
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MATHEMATICA
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f[x_] = (1/(48*(1 + ProductLog[-x])^6))* ProductLog[-x]^4*(2 - 28*ProductLog[-x] - 23*ProductLog[-x]^2 - 9*ProductLog[-x]^3 - ProductLog[-x]^4); Rest[CoefficientList[Series[f[x], {x, 0, 17}], x]*Range[0, 17]!] (* Jean-François Alcover, Jul 11 2011, after formula *)
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PROG
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(PARI) N=66; x='x+O('x^N); /* that many terms */
T=sum(n=1, N, n^(n-1)/n!*x^n); /* e.g.f. of A000169 */
egf=1/48*T^4*(2+28*T-23*T^2+9*T^3-T^4)/(1-T)^6;
Vec(serlaplace(egf)) /* show terms, starting with 1 */
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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RAVELOMANANA Vlady (vlad(AT)lri.fr), May 16 2001
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STATUS
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approved
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