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 A007719 Number of independent polynomial invariants of symmetric matrix of order n. 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 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. FORMULA Inverse Euler transform of A007717. EXAMPLE 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) 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 *) CROSSREFS Row sums of A322115. Cf. A002905, A007716, A007717, A007719, A020555, A050535, A053419, A076864, A191970, A316972, A316974. Sequence in context: A148153 A148154 A148155 * A148156 A148157 A141268 Adjacent sequences:  A007716 A007717 A007718 * A007720 A007721 A007722 KEYWORD nonn,nice AUTHOR EXTENSIONS a(0)=1 added by Alberto Tacchella, Jun 20 2011 a(7)-a(25) from Franklin T. Adams-Watters, Jun 21 2011 STATUS approved

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Last modified August 3 11:42 EDT 2020. Contains 336198 sequences. (Running on oeis4.)