A095340 Total number of nodes in all labeled graphs on n nodes.

1, 4, 24, 256, 5120, 196608, 14680064, 2147483648, 618475290624, 351843720888320, 396316767208603648, 885443715538058477568, 3929008913747544817795072, 34662321099990647697175478272, 608472288109550112718417538580480, 21267647932558653966460912964485513216

Offset: 1

Author

Eric W. Weisstein, Jun 03 2004

Keywords

• nonn

Comments

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

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..50
• Mihai Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
• Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part I, Journal of Algebraic Combinatorics 1-2, 111-132 (1992).
• Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part II, Journal of Algebraic Combinatorics 1-3, 219-234 (1992).
• Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
• Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, Volume 6 (1999), Research Paper #R16.
• Christian Krattenthaler, Schur function identities and the number of perfect matchings of Aztec holey rectangles, arXiv:math/9712204 [math.CO], 1997.
• 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, Solving graph problems using permutation-invariant quantum machine learning, arXiv:2505.12764 [quant-ph], 2025. See p. 9.
• Eric Weisstein's World of Mathematics, Graph Vertex

Crossrefs

Cf. A006125, A103904.

Programs

Magma

[n*2^((n^2-n) div 2): n in [1..20]]; // Vincenzo Librandi, Aug 17 2015

Maple

a:= n-> n*2^(n*(n-1)/2):
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 26 2013

Mathematica

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 *)

PARI

a(n)=n*2^((n^2-n)/2)

Formula

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

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

a(n) = (32*a(n-1)*a(n-3)-48*a(n-2)^2)/a(n-4). - Michael Somos, Sep 16 2005

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

Extensions

Edited by Ralf Stephan, Feb 21 2005