A241839 Number of simple connected graphs on n nodes that are not regular.

0, 0, 1, 4, 19, 107, 849, 11100, 261058, 11716404, 1006700026, 164059811497, 50335907479783, 29003487412533265, 31397381139819043520, 63969560111526659139866, 245871831681641239553413008, 1787331725248249110678608976294, 24636021429399437942454151113206764

Offset: 1

Author

Travis Hoppe and Anna Petrone, Apr 29 2014

Keywords

• nonn

Links

• Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
• T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
• Eric Weisstein's World of Mathematics, Regular Graph

Crossrefs

Cf. A002851, A006820, A006822.

Programs

Mathematica

A005177 = {1, 1, 1, 1, 2, 2, 5, 4, 17, 22, 167, 539, 18979, 389436, 50314796, 2942198440, 1698517036411};
terms = Length[A005177] - 1;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
A001349 = Join[{1}, EULERi[Array[a88, terms]]];
Rest[A001349 - A005177] (* Jean-François Alcover, Feb 23 2019, after Andrew Howroyd *)

Formula

a(n) = A001349(n) - A005177(n). - Andrew Howroyd, Nov 04 2017

Extensions

a(11)-a(16) from Andrew Howroyd, Nov 04 2017

Terms a(17) and beyond from Andrew Howroyd, May 21 2020