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 A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes. 76
 1, 2, 7, 23, 79, 274, 1003, 3763, 14723, 59663, 250738, 1090608, 4905430, 22777420, 109040012, 537401702, 2723210617, 14170838544, 75639280146, 413692111521, 2316122210804, 13261980807830, 77598959094772, 463626704130058, 2826406013488180, 17569700716557737 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Euler transform of A007719. Also the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018 Number of distinct n X 2n matrices with integer entries and rows sums 2, up to row and column permutations. - Andrew Howroyd, Sep 06 2018 a(n) is the number of unlabeled multigraphs with n edges rooted at one vertex. - Andrew Howroyd, Nov 22 2020 REFERENCES Huaien Li and David C. Torney, Enumerations of Multigraphs, 2002. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..50 Huaien Li and David C. Torney, Enumeration of unlabelled multigraphs, Ars Combin. 75 (2005) 171-188. MR2133219. R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) table 67. EXAMPLE a(2) = 7 (here - denotes an edge, = denotes a pair of parallel edges and o is a loop): oo o o o- o - = -- - - From Gus Wiseman, Jul 18 2018: (Start) Non-isomorphic representatives of the a(2) = 7 multiset partitions of {1, 1, 2, 2}:   (1122),   (1)(122), (11)(22), (12)(12),   (1)(1)(22), (1)(2)(12),   (1)(1)(2)(2). (End) MATHEMATICA 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!]; a[n_] := RowSumMats[n, 2n, 2]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *) PROG (PARI) \\ See A318951 for RowSumMats a(n)=RowSumMats(n, 2*n, 2); \\ Andrew Howroyd, Sep 06 2018 (PARI) \\ See A339065 for G. seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, ))} \\ Andrew Howroyd, Nov 22 2020 CROSSREFS Row n=2 of A331485. Cf. A000664, A002620, A007716, A007719, A020555, A050531, A050532, A050535, A052171, A053418, A053419, A094574, A316972, A316974, A318951, A339065. Sequence in context: A198944 A328035 A112657 * A130567 A091514 A143629 Adjacent sequences:  A007714 A007715 A007716 * A007718 A007719 A007720 KEYWORD nonn AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Jan 26 2000 a(0)=1 prepended and a(16)-a(25) added by Max Alekseyev, Jun 21 2011 STATUS approved

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Last modified September 24 18:11 EDT 2022. Contains 356949 sequences. (Running on oeis4.)