OFFSET
0,3
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 80, Eq. 3.11.5.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..50
Vladislav Bína and Jiří Přibil, Note on enumeration of labeled split graphs, Comment. Math. Univ. Carolin. 56,2 (2015) 133-137.
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.
S. R. Finch, Bipartite, k-colorable and k-colored graphs. [Broken link]
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 32.
Eric Weisstein's World of Mathematics, n-Colorable Graph.
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 89, Eq. 3.11.5.
FORMULA
E.g.f.: sqrt( e.g.f. for A047863 ).
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. - Mehdi Naima, Jun 05 2026
MATHEMATICA
nn = 20; a = Sum[Sum[Binomial[n, k] 2^(k (n - k)), {k, 0, n}] x^n/n!, {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[a^(1/2), {x, 0, nn}], x] (* Geoffrey Critzer, Jan 15 2012 *)
PROG
(PARI) N=18; x='x+O('x^N); Vec(serlaplace(sqrt(sum(n=0, N, exp(2^n*x)*x^n/n!)))) \\ Gheorghe Coserea, Nov 13 2017
CROSSREFS
KEYWORD
nonn,nice,easy,changed
AUTHOR
STATUS
approved