This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A095340 #48 May 29 2025 11:09:13
%S A095340 1,4,24,256,5120,196608,14680064,2147483648,618475290624,
%T A095340 351843720888320,396316767208603648,885443715538058477568,
%U A095340 3929008913747544817795072,34662321099990647697175478272,608472288109550112718417538580480,21267647932558653966460912964485513216
%N A095340 Total number of nodes in all labeled graphs on n nodes.
%C A095340 Number of perfect matchings of an n X (n+1) Aztec rectangle with the second vertex in the topmost row removed.
%D A095340 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7
%H A095340 Alois P. Heinz, <a href="/A095340/b095340.txt">Table of n, a(n) for n = 1..50</a>
%H A095340 Mihai Ciucu, <a href="https://mciucu.pages.iu.edu/symmgraphs.pdf">Enumeration of perfect matchings in graphs with reflective symmetry</a>, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
%H A095340 Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, <a href="http://dx.doi.org/10.1023/A:1022420103267">Alternating sign matrices and domino tilings. Part I</a>, Journal of Algebraic Combinatorics 1-2, 111-132 (1992).
%H A095340 Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, <a href="http://dx.doi.org/10.1023/A:1022483817303">Alternating sign matrices and domino tilings. Part II</a>, Journal of Algebraic Combinatorics 1-3, 219-234 (1992).
%H A095340 Harald Helfgott and Ira M. Gessel, <a href="http://arxiv.org/abs/math/9810143">Enumeration of tilings of diamonds and hexagons with defects</a>, arXiv:math/9810143 [math.CO], 1998.
%H A095340 Harald Helfgott and Ira M. Gessel, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v6i1r16/0">Enumeration of tilings of diamonds and hexagons with defects</a>, Volume 6 (1999), Research Paper #R16.
%H A095340 Christian Krattenthaler, <a href="http://arxiv.org/abs/math/9712204">Schur function identities and the number of perfect matchings of Aztec holey rectangles</a>, arXiv:math/9712204 [math.CO], 1997.
%H A095340 Maximilian Balthasar Mansky, Tobias Rohe, Gerhard Stenzel, Alejandro Bravo de la Serna, Santiago Londoño Castillo, Gautham Sathish, Dimitra Nikolaidou, Dmytro Bondarenko, Linus Menzel, and Claudia Linnhoff-Popien, <a href="https://arxiv.org/abs/2505.12764">Solving graph problems using permutation-invariant quantum machine learning</a>, arXiv:2505.12764 [quant-ph], 2025. See p. 9.
%H A095340 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphVertex.html">Graph Vertex</a>
%F A095340 a(n) = n * A006125(n).
%F A095340 a(n) = n * 2^(n*(n-1)/2). E.g., a(7) = 7 * 2^(7*6/2) = 7 * 2097152 = 14680064. - _David Terr_, Nov 08 2004
%F A095340 a(n) = (32*a(n-1)*a(n-3)-48*a(n-2)^2)/a(n-4). - _Michael Somos_, Sep 16 2005
%F A095340 a(n) = Sum_{k=1..n} k*C(n,k)*2^C(n-k,2)*A001187(k). The sum gives the number of rooted labeled simple graphs on n nodes. It conditions on k, 1<=k<=n, the size of the connected component that the root is in. See Harary and Palmer reference. - _Geoffrey Critzer_, Nov 12 2011
%p A095340 a:= n-> n*2^(n*(n-1)/2):
%p A095340 seq(a(n), n=1..20); # _Alois P. Heinz_, Aug 26 2013
%t A095340 g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, 20}]; a = Drop[Range[0, 20]! CoefficientList[Series[Log[g] + 1, {x, 0, 20}], x], 1]; Table[Sum[k Binomial[n, k] 2^Binomial[n - k, 2] a[[k]], {k, 1, n}], {n, 1,20}] (* _Geoffrey Critzer_, Nov 12 2011 *)
%o A095340 (PARI) a(n)=n*2^((n^2-n)/2)
%o A095340 (Magma) [n*2^((n^2-n) div 2): n in [1..20]]; // _Vincenzo Librandi_, Aug 17 2015
%Y A095340 Cf. A006125, A103904.
%K A095340 nonn
%O A095340 1,2
%A A095340 _Eric W. Weisstein_, Jun 03 2004
%E A095340 Edited by _Ralf Stephan_, Feb 21 2005