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Number of simple connected graphs on n nodes that are not regular.
2

%I #21 May 21 2020 12:52:46

%S 0,0,1,4,19,107,849,11100,261058,11716404,1006700026,164059811497,

%T 50335907479783,29003487412533265,31397381139819043520,

%U 63969560111526659139866,245871831681641239553413008,1787331725248249110678608976294,24636021429399437942454151113206764

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

%H Travis Hoppe and Anna Petrone, <a href="https://github.com/thoppe/Encyclopedia-of-Finite-Graphs">Encyclopedia of Finite Graphs</a>

%H T. Hoppe and A. Petrone, <a href="http://arxiv.org/abs/1408.3644">Integer sequence discovery from small graphs</a>, arXiv preprint arXiv:1408.3644 [math.CO], 2014.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RegularGraph.html">Regular Graph</a>

%F a(n) = A001349(n) - A005177(n). - _Andrew Howroyd_, Nov 04 2017

%t A005177 = {1, 1, 1, 1, 2, 2, 5, 4, 17, 22, 167, 539, 18979, 389436, 50314796, 2942198440, 1698517036411};

%t terms = Length[A005177] - 1;

%t mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];

%t 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]];

%t 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];

%t edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];

%t a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];

%t A001349 = Join[{1}, EULERi[Array[a88, terms]]];

%t Rest[A001349 - A005177] (* _Jean-François Alcover_, Feb 23 2019, after _Andrew Howroyd_ *)

%Y Cf. A002851, A006820, A006822.

%K nonn

%O 1,4

%A _Travis Hoppe_ and _Anna Petrone_, Apr 29 2014

%E a(11)-a(16) from _Andrew Howroyd_, Nov 04 2017

%E Terms a(17) and beyond from _Andrew Howroyd_, May 21 2020