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%I A007717 #62 Jan 08 2024 20:59:25
%S A007717 1,2,7,23,79,274,1003,3763,14723,59663,250738,1090608,4905430,
%T A007717 22777420,109040012,537401702,2723210617,14170838544,75639280146,
%U A007717 413692111521,2316122210804,13261980807830,77598959094772,463626704130058,2826406013488180,17569700716557737
%N 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.
%C A007717 Euler transform of A007719.
%C A007717 Also the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - _Gus Wiseman_, Jul 18 2018
%C A007717 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
%C A007717 a(n) is the number of unlabeled loopless multigraphs with n edges rooted at one vertex. - _Andrew Howroyd_, Nov 22 2020
%D A007717 Huaien Li and David C. Torney, Enumerations of Multigraphs, 2002.
%H A007717 Andrew Howroyd, <a href="/A007717/b007717.txt">Table of n, a(n) for n = 0..50</a>
%H A007717 Huaien Li and David C. Torney, Enumeration of unlabelled multigraphs, Ars Combin. 75 (2005) 171-188. <a href="http://www.ams.org/mathscinet-getitem?mr=2133219">MR2133219</a>.
%H A007717 R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO] (2017) table 67.
%e A007717 a(2) = 7 (here - denotes an edge, = denotes a pair of parallel edges and o is a loop):
%e A007717 oo
%e A007717 o o
%e A007717 o-
%e A007717 o -
%e A007717 =
%e A007717 --
%e A007717 - -
%e A007717 From _Gus Wiseman_, Jul 18 2018: (Start)
%e A007717 Non-isomorphic representatives of the a(2) = 7 multiset partitions of {1, 1, 2, 2}:
%e A007717   (1122),
%e A007717   (1)(122), (11)(22), (12)(12),
%e A007717   (1)(1)(22), (1)(2)(12),
%e A007717   (1)(1)(2)(2).
%e A007717 (End)
%e A007717 From _Gus Wiseman_, Jan 08 2024: (Start)
%e A007717 Non-isomorphic representatives of the a(1) = 1 through a(3) = 7 rooted loopless multigraphs (root shown as singleton):
%e A007717   {{1}}  {{1},{1,2}}  {{1},{1,2},{1,2}}
%e A007717          {{1},{2,3}}  {{1},{1,2},{1,3}}
%e A007717                       {{1},{1,2},{2,3}}
%e A007717                       {{1},{1,2},{3,4}}
%e A007717                       {{1},{2,3},{2,3}}
%e A007717                       {{1},{2,3},{2,4}}
%e A007717                       {{1},{2,3},{4,5}}
%e A007717 (End)
%t A007717 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 A007717 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 A007717 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 A007717 a[n_] := RowSumMats[n, 2n, 2];
%t A007717 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* _Jean-François Alcover_, Oct 27 2018, after _Andrew Howroyd_ *)
%o A007717 (PARI) \\ See A318951 for RowSumMats
%o A007717 a(n)=RowSumMats(n, 2*n, 2); \\ _Andrew Howroyd_, Sep 06 2018
%o A007717 (PARI) \\ See A339065 for G.
%o A007717 seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1]))} \\ _Andrew Howroyd_, Nov 22 2020
%Y A007717 Row n=2 of A331485.
%Y A007717 Cf. A000664, A002620, A007716, A007719, A020555, A050531, A050532, A050535, A052171, A053418, A053419, A094574, A316972, A316974, A318951, A339065.
%K A007717 nonn
%O A007717 0,2
%A A007717 _Colin Mallows_
%E A007717 More terms from _Vladeta Jovovic_, Jan 26 2000
%E A007717 a(0)=1 prepended and a(16)-a(25) added by _Max Alekseyev_, Jun 21 2011

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