A061544 Number of connected labeled graphs with n nodes and n+6 edges.
0, 0, 0, 0, 0, 455, 202755, 39183840, 5228627544, 573177986865, 56169415897650, 5157436533796140, 456501786661617840, 39667302684866008152, 3425100498297691978050, 296331952661358892037760, 25839208713048103250144280, 2280203173608621371757204480, 204244225852123476144894896712
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..380
- Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
- S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, arXiv:math/9310236 [math.PR], 1993.
- S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
- E. M. Wright, The Number of Connected Sparsely Edged Graphs, Journal of Graph Theory Vol. 1 (1977), 317-330.
Programs
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Mathematica
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 *)
Formula
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)).