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%I #19 Apr 16 2021 15:43:05
%S 0,0,0,0,0,455,202755,39183840,5228627544,573177986865,56169415897650,
%T 5157436533796140,456501786661617840,39667302684866008152,
%U 3425100498297691978050,296331952661358892037760
%N Number of connected labeled graphs with n nodes and n+6 edges.
%H G. C. Greubel, <a href="/A061544/b061544.txt">Table of n, a(n) for n = 1..380</a>
%H S. Janson, D. E. Knuth, T. Luczak and B. Pittel, <a href="http://arxiv.org/abs/math/9310236">The Birth of the Giant Component</a>, arXiv:math/9310236 [math.PR], 1993.
%H S. Janson, D. E. Knuth, T. Luczak and B. Pittel, <a href="http://dx.doi.org/10.1002/rsa.3240040303">The Birth of the Giant Component</a>, Random Structures and Algorithms Vol. 4 (1993), 233-358.
%H E. M. Wright, <a href="http://dx.doi.org/10.1002/jgt.3190020403">The Number of Connected Sparsely Edged Graphs</a>, Journal of Graph Theory Vol. 1 (1977), 317-330.
%F E.g.f.: W6(x) = - 1/5806080*T(x)^6*( - 3669120 - 145514880*T(x) - 826813440*T(x)^2 - 160242624*T(x)^3 + 549065304*T(x)^4 - 1423242144*T(x)^5 + 1649073392*T(x)^6 - 1408032768*T(x)^7 + 881917344*T(x)^8 - 418233349*T(x)^9 + 147585749*T(x)^10 - 37755372*T(x)^11 + 6581528*T(x)^12 - 696620*T(x)^13 + 33000*T(x)^14)/(( - 1 + T(x))^18) where T(x) is the e.g.f. for rooted labeled trees (A000169), i.e., T(x) = -LambertW(-x) = x*exp(T(x)).
%t t[x_] := -ProductLog[-x]; W6[x_] := -1/5806080*t[x]^6*(-3669120 - 145514880*t[x] - 826813440*t[x]^2 - 160242624*t[x]^3 + 549065304*t[x]^4 - 1423242144*t[x]^5 + 1649073392*t[x]^6 - 1408032768*t[x]^7 + 881917344*t[x]^8 - 418233349*t[x]^9 + 147585749*t[x]^10 - 37755372*t[x]^11 + 6581528*t[x]^12 - 696620*t[x]^13 +33000*t[x]^14)/((-1 + t[x])^18); max = 20; CoefficientList[Series[W6[x], {x, 0, max}], x]*Range[0, max]! // Rest (* _G. C. Greubel_, Nov 12 2017 *)
%Y A diagonal of A343088.
%Y Cf. A000169, A000272, A057500.
%K easy,nice,nonn
%O 1,6
%A Ravelomanana Vlady (vlad(AT)lri.fr), May 16 2001
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