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%I #53 Mar 22 2024 11:06:02
%S 1,1,3,8,23,66,212,686,2389,8682,33160,132277,550835,2384411,10709827,
%T 49782637,238998910,1182772364,6023860266,31525780044,169316000494,
%U 932078457785,5253664040426,30290320077851,178480713438362,1073918172017297
%N Number of loopless multigraphs on infinite set of nodes with n edges.
%C Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - _Max Alekseyev_, Oct 18 2017
%C Row sums of the triangle defined by the Multiset Transformation of A076864,
%C 1 ;
%C 0 1;
%C 0 2 1;
%C 0 5 2 1;
%C 0 12 8 2 1;
%C 0 33 22 8 2 1;
%C 0 103 72 26 8 2 1;
%C 0 333 229 87 26 8 2 1;
%C 0 1183 782 295 92 26 8 2 1;
%C 0 4442 2760 1036 315 92 26 8 2 1;
%C 0 17576 10270 3735 1129 321 92 26 8 2 1;
%C 0 72810 39770 13976 4117 1154 321 92 26 8 2 1;
%C 0 314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;
%C - _R. J. Mathar_, Jul 18 2017
%C Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - _Gus Wiseman_, Jul 18 2018
%D Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).
%H Andrew Howroyd, <a href="/A050535/b050535.txt">Table of n, a(n) for n = 0..50</a>
%H George Barnes, Sanjaye Ramgoolam, and Michael Stephanou, <a href="https://arxiv.org/abs/2306.04569">Permutation invariant Gaussian matrix models for financial correlation matrices</a>, arXiv:2306.04569 [q-fin.ST], 2023.
%H Frank Harary, <a href="http://dx.doi.org/10.1090/S0002-9947-1955-0068198-2">The number of linear, directed, rooted, and connected graphs</a>, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
%H Vladeta Jovovic, <a href="/A050535/a050535.pdf">Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation</a>
%H Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, <a href="https://arxiv.org/abs/1712.07124">Energy flow polynomials: A complete linear basis for jet substructure</a>, arXiv:1712.07124 [hep-ph], 2017.
%H Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, <a href="https://arxiv.org/abs/2403.09985">Universal graph series, chromatic functions, and their index theory</a>, arXiv:2403.09985 [math.CO], 2024. See p. 23.
%F a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - _Max Alekseyev_, Oct 18 2017
%F Euler transform of A076864. - _Andrew Howroyd_, Oct 23 2019
%e From _Gus Wiseman_, Jul 18 2018: (Start)
%e Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
%e (123)(123)
%e (1)(23)(123)
%e (12)(13)(23)
%e (1)(1)(23)(23)
%e (1)(2)(3)(123)
%e (1)(2)(13)(23)
%e (1)(1)(2)(3)(23)
%e (1)(1)(2)(2)(3)(3)
%e (End)
%t seq[n_] := G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
%t seq[15] (* _Jean-François Alcover_, Dec 02 2020, using _Andrew Howroyd_'s code for G in A339065 *)
%Y Cf. A001399, A003082, A014395, A014396, A014397, A014398.
%Y Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.
%Y Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.
%K nonn
%O 0,3
%A _Vladeta Jovovic_, Dec 29 1999
%E More terms from _Sean A. Irvine_, Oct 02 2011
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