A004100 Number of labeled nonseparable bipartite graphs on n nodes.

0, 1, 0, 3, 10, 355, 6986, 297619, 15077658, 1120452771, 111765799882, 15350524923547, 2875055248515242, 738416821509929731, 260316039943139322858, 126430202628042630866787, 84814075550928212558332858, 78847417416749666369637926851

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100 (terms 1..32 from R. W. Robinson)
• 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)
• 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of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]

Crossrefs

Cf. A001832, A013922.

Programs

Mathematica

b[n_] := Log[Sum[Exp[2^k*x + O[x]^n]*x^k/k!, {k, 0, n}]/2];
seq[n_] := CoefficientList[-Log[2] + Log[x/InverseSeries[x*D[b[n], x]]], x]*Table[(2k)!!, {k, 0, n-2}];
seq[19] (* Jean-François Alcover, Sep 04 2019, after Andrew Howroyd *)

PARI

\\ here b(n) is A001832 as e.g.f.
b(n)={log(sum(k=0, n, exp(2^k*x + O(x*x^n))*x^k/k!))/2}
seq(n)={Vec(serlaplace(log(x/serreverse(x*deriv(b(n))))), -n)} \\ Andrew Howroyd, Sep 26 2018

Extensions

a(16) onwards added by N. J. A. Sloane, Oct 19 2006 from the Robinson reference