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Total number of nodes in all labeled graphs on n nodes.
10

%I #43 Sep 08 2022 08:45:13

%S 1,4,24,256,5120,196608,14680064,2147483648,618475290624,

%T 351843720888320,396316767208603648,885443715538058477568,

%U 3929008913747544817795072,34662321099990647697175478272,608472288109550112718417538580480,21267647932558653966460912964485513216

%N Total number of nodes in all labeled graphs on n nodes.

%C Number of perfect matchings of an n X (n+1) Aztec rectangle with the second vertex in the topmost row removed.

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7

%H Alois P. Heinz, <a href="/A095340/b095340.txt">Table of n, a(n) for n = 1..50</a>

%H M. 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 N. Elkies, G. Kuperberg, M. Larsen and J. 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 N. Elkies, G. Kuperberg, M. Larsen and J. 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 H. Helfgott and I. 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 H. Helfgott and I. 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 C. 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphVertex.html">Graph Vertex</a>

%F a(n) = n * A006125(n).

%F 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 a(n) = (32*a(n-1)*a(n-3)-48*a(n-2)^2)/a(n-4). - _Michael Somos_, Sep 16 2005

%F 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 a:= n-> n*2^(n*(n-1)/2):

%p seq(a(n), n=1..20); # _Alois P. Heinz_, Aug 26 2013

%t 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 (PARI) a(n)=n*2^((n^2-n)/2)

%o (Magma) [n*2^((n^2-n) div 2): n in [1..20]]; // _Vincenzo Librandi_, Aug 17 2015

%Y Cf. A006125, A103904.

%K nonn

%O 1,2

%A _Eric W. Weisstein_, Jun 03 2004

%E Edited by _Ralf Stephan_, Feb 21 2005