OFFSET
1,3
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..50
Simon Dreyer, Antoine Genitrini, and Mehdi Naima, Asymptotic Enumeration of Labeled Triangle-Free Graphs through the Combinatorics of Directed Acyclic Graphs of Shortest Paths, hal:05609965, 2026.
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy)
David A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
David A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
A. Nymeyer and R. W. Robinson, Tabulation of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]
Kai Wang, An Efficient Algorithm to Generate all Labeled Triangle-free Graphs with a given Graphical Degree Sequence, arXiv:2601.15943 [math.CO], 2026. See p. 5.
Eric Weisstein's World of Mathematics, n-Colorable Graph
Eric Weisstein's World of Mathematics, n-Chromatic Graph
FORMULA
E.g.f.: log(A(x))/2 where A(x) is e.g.f. of A047863.
a(n) = A002031(n)/2, for n > 1. - Geoffrey Critzer, May 10 2011
From Mehdi Naima, Jun 05 2026: (Start)
a(0) = 1, a(n) = Sum_{k = 1..n} r(n,k), r(n,k) = 1 if n = k and r(n,k) = binomial(n,k)*Sum_{l = 1..n-k} (2^l - 1)^k*r(n-k,l) otherwise.
a(n) ~ c * 2^(n^2/4+n-1/2)/sqrt(Pi*n), where c = Sum_{k = -oo..oo} 2^(-k^2) = EllipticTheta[3, 0, 1/2] = 2.128936827211877... if n is even and c = Sum_{k = -oo..oo} 2^(-(k+1/2)^2) = EllipticTheta[2, 0, 1/2] = 2.12893125051302... if n is odd. (End)
MATHEMATICA
mx = 17; s = Sum[ Binomial[n, k] 2^(k (n - k)) x^n/n!, {n, 0, mx}, {k, 0, n}] ; Range[0, mx]! CoefficientList[ Series[ Log[s]/2, {x, 0, mx}], x] (* Geoffrey Critzer, May 10 2011 *)
PROG
(PARI) seq(n)=Vec(serlaplace(log(sum(k=0, n, exp(2^k*x + O(x*x^n))*x^k/k!))/2)) \\ Andrew Howroyd, Sep 26 2018
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 12 2003
STATUS
approved