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

1, 2, 4, 11, 30, 95, 328, 1211, 4779, 19902, 86682, 393072, 1847264, 8965027, 44814034, 230232789, 1213534723, 6552995689, 36207886517, 204499421849, 1179555353219, 6942908667578, 41673453738272, 254918441681030, 1588256152307002, 10073760672179505

Offset: 0

Author

Colin Mallows

Keywords

• nonn
• nice

Comments

Also, number of connected multigraphs with n edges (allowing loops) and any number of nodes.

Also the number of non-isomorphic connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

Examples

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 11 connected multiset partitions of {1, 1, 2, 2, 3, 3}:
  (112233),
  (1)(12233), (12)(1233), (112)(233), (123)(123),
  (1)(2)(1233), (1)(12)(233), (1)(23)(123), (12)(13)(23),
  (1)(2)(3)(123), (1)(2)(13)(23).
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 63.

Crossrefs

Row sums of A322115.

Cf. A002905, A007716, A007717, A007719, A020555, A050535, A053419, A076864, A191970, A316972, A316974.

Programs

Mathematica

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];
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}];
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!];
A007717 = Table[Print[n]; RowSumMats[n, 2 n, 2], {n, 0, 20}];
Join[{1}, EULERi[Rest[A007717]]] (* Jean-François Alcover, Oct 29 2018, using Andrew Howroyd's code for A007717 *)

Formula

Inverse Euler transform of A007717.

Extensions

a(0)=1 added by Alberto Tacchella, Jun 20 2011

a(7)-a(25) from Franklin T. Adams-Watters, Jun 21 2011