A001832 Number of labeled connected bipartite graphs on n nodes. (Formerly M3063 N1241)

1, 1, 3, 19, 195, 3031, 67263, 2086099, 89224635, 5254054111, 426609529863, 47982981969979, 7507894696005795, 1641072554263066471, 502596525992239961103, 216218525837808950623459, 130887167385831881114006475, 111653218763166828863141636911

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

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)

D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.

D. 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of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]

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

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

Row sums of A228861.

The unlabeled version is A005142.

Cf. A002031, A047863, A047864.

Sequence in context: A048172 A079145 A000763 * A195511 A123681 A007151

Adjacent sequences: A001829 A001830 A001831 * A001833 A001834 A001835

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 12 2003

Status

approved