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A007719 Number of independent polynomial invariants of symmetric matrix of order n.

%I A007719 #39 Oct 23 2019 14:35:38
%S A007719 1,2,4,11,30,95,328,1211,4779,19902,86682,393072,1847264,8965027,
%T A007719 44814034,230232789,1213534723,6552995689,36207886517,204499421849,
%U A007719 1179555353219,6942908667578,41673453738272,254918441681030,1588256152307002,10073760672179505
%N A007719 Number of independent polynomial invariants of symmetric matrix of order n.
%C A007719 Also, number of connected multigraphs with n edges (allowing loops) and any number of nodes.
%C A007719 Also the number of non-isomorphic connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - _Gus Wiseman_, Jul 18 2018
%H A007719 Andrew Howroyd, <a href="/A007719/b007719.txt">Table of n, a(n) for n = 0..50</a>
%H A007719 R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO] (2017) Table 63.
%F A007719 Inverse Euler transform of A007717.
%e A007719 From _Gus Wiseman_, Jul 18 2018: (Start)
%e A007719 Non-isomorphic representatives of the a(3) = 11 connected multiset partitions of {1, 1, 2, 2, 3, 3}:
%e A007719   (112233),
%e A007719   (1)(12233), (12)(1233), (112)(233), (123)(123),
%e A007719   (1)(2)(1233), (1)(12)(233), (1)(23)(123), (12)(13)(23),
%e A007719   (1)(2)(3)(123), (1)(2)(13)(23).
%e A007719 (End)
%t A007719 mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t A007719 EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++,
%t A007719   c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {};
%t A007719   For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
%t A007719 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 A007719 Kq[q_, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
%t A007719 RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[ Exp[Sum[Kq[q, t, k]/t x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
%t A007719 A007717 = Table[Print[n]; RowSumMats[n, 2 n, 2], {n, 0, 20}];
%t A007719 Join[{1}, EULERi[Rest[A007717]]] (* _Jean-François Alcover_, Oct 29 2018, using _Andrew Howroyd_'s code for A007717 *)
%Y A007719 Row sums of A322115.
%Y A007719 Cf. A002905, A007716, A007717, A007719, A020555, A050535, A053419, A076864, A191970, A316972, A316974.
%K A007719 nonn,nice
%O A007719 0,2
%A A007719 _Colin Mallows_
%E A007719 a(0)=1 added by _Alberto Tacchella_, Jun 20 2011
%E A007719 a(7)-a(25) from _Franklin T. Adams-Watters_, Jun 21 2011