This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A061543 #30 Jun 22 2025 10:15:24
%S A061543 0,0,0,0,10,2997,343140,28044072,1969994376,128916045720,
%T A061543 8189607254829,516895556463000,32865110582830812,2123144102136625568,
%U A061543 140115162250240202025,9478591551140049252096,658706750876277003711720,47086655712339052407435264,3464805563040942592258054518
%N A061543 Number of connected labeled graphs with n nodes and n+4 edges.
%H A061543 G. C. Greubel, <a href="/A061543/b061543.txt">Table of n, a(n) for n = 1..382</a>
%H A061543 Steven R. Finch, <a href="https://arxiv.org/abs/2408.12440">An exceptional convolutional recurrence</a>, arXiv:2408.12440 [math.CO], 22 Aug 2024.
%H A061543 S. Janson, D. E. Knuth, T. Łuczak 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 A061543 S. Janson, D. E. Knuth, T. Łuczak 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 A061543 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 A061543 E.g.f.: W4(x) = - 1/11520*T(x)^5*( - 960 - 31632*T(x) - 54144*T(x)^2 + 100976*T(x)^3 - 117368*T(x)^4 + 79520*T(x)^5 - 35793*T(x)^6 + 10069*T(x)^7 - 1626*T(x)^8 + 108*T(x)^9)/(( - 1 + T(x))^12) where T(x) is the e.g.f. for rooted labeled trees (A000169), i.e., T(x) = - LambertW( - x) = x*exp(T(x)).
%t A061543 max=17; t[x_] := -ProductLog[-x]; w4[x_] := -1/11520*t[x]^5*(-960 - 31632*t[x] - 54144*t[x]^2 + 100976*t[x]^3 - 117368*t[x]^4 + 79520*t[x]^5 - 35793*t[x]^6 + 10069*t[x]^7 - 1626*t[x]^8 + 108*t[x]^9) / (-1 + t[x])^12; CoefficientList[ Series[w4[x], {x, 0, max}], x]*Range[0, max]! // Rest (* _Jean-François Alcover_, Sep 07 2012, from e.g.f. *)
%Y A061543 A diagonal of A343088.
%Y A061543 Cf. A000169, A000272.
%K A061543 easy,nice,nonn
%O A061543 1,5
%A A061543 Ravelomanana Vlady (vlad(AT)lri.fr), May 16 2001