A047864 Number of labeled bipartite graphs with n nodes.

1, 1, 2, 7, 41, 376, 5177, 103237, 2922446, 116011231, 6433447397, 498234407452, 54007795331921, 8213123246906761, 1756336596363006842, 528975889250504033527, 224688018516023267969441, 134708289561117007261966816

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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.
• 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.

Crossrefs

Row sums of A117279.

The unlabeled version is A033995.

Cf. A001832, A047863.

Programs

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 *)

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

Formula

E.g.f.: sqrt( e.g.f. for A047863 ).