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A061541 Number of connected labeled graphs with n nodes and n+2 edges. 9
0, 0, 0, 1, 120, 6165, 258125, 10230360, 405918324, 16530124800, 699126562530, 30884683104000, 1428626760992860, 69248819808744576, 3516693960681822375, 186964957159176734720, 10395215954531344335000, 603712553730550509035520, 36575888366817680447745924 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 1..380 (first 100 terms from T. D. Noe)

S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.

S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The Birth of the Giant Component, arXiv:math/9310236 [math.PR], 1993.

E. M. Wright, The Number of Connected Sparsely Edged Graphs, Journal of Graph Theory Vol. 1 (1977), 317-330.

FORMULA

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

MATHEMATICA

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-Fran├žois Alcover, Jul 11 2011, after formula *)

PROG

(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 */

/* Joerg Arndt, Jul 11 2011 */

CROSSREFS

Cf. A000169, A000272, A057500, A061541, A061542, etc.

Sequence in context: A246216 A246284 A289292 * A003438 A222157 A092710

Adjacent sequences:  A061538 A061539 A061540 * A061542 A061543 A061544

KEYWORD

easy,nice,nonn

AUTHOR

RAVELOMANANA Vlady (vlad(AT)lri.fr), May 16 2001

STATUS

approved

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Last modified December 7 20:25 EST 2019. Contains 329848 sequences. (Running on oeis4.)