A297877 -id:A297877 - cp's OEIS frontend

A033995 Number of bipartite graphs with n nodes.

1, 1, 2, 3, 7, 13, 35, 88, 303, 1119, 5479, 32303, 251135, 2527712, 33985853, 611846940, 14864650924, 488222721992, 21712049275198, 1308300679611469, 106897965189674291, 11852113048215107822, 1784730721403509209215, 365323537513403184463273

Offset: 0

Ronald C. Read

All bipartite graphs are perfect. - Falk Hüffner, Nov 27 2015

EULER transform of A005142 [1, 1, 1, 3, 5, 17, ...] is [1, 2, 3, 7, 13, ...]. - Michael Somos, May 13 2019

			For n=1: o; n=2: o o, o-o; n=3: o o o, o o-o, o-o-o; n=4: o o o o, o o o-o, o-o o-o, o o-o-o, o-o-o-o, K_{2,2}, K_{3,1}. - _Michael Somos_, May 13 2019

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

Andrew Howroyd, Table of n, a(n) for n = 0..50
CombOS - Combinatorial Object Server, Generate graphs.
P. Erdős, D. J. Kleitman, and B. L. Rothschild, Asymptotic enumeration of k_n-free graphs. In Colloquio Internazionale sulle Teorie Combinatorie, (Rome, 1973), Tomo II, Atti dei Convegni Lincei, No. 17, pp. 19-27. Accad. Naz. Lincei, Rome.
P. Hanlon, The enumeration of bipartite graphs, Discrete Math. 28 (1979), 49-57.
S. Hougardy, Home Page.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
Sage, Common Graphs (Graph Generators).
Eric Weisstein's World of Mathematics, Bipartite Graph.
Eric Weisstein's World of Mathematics, Bicolorable Graph.
Eric Weisstein's World of Mathematics, n-Colorable Graph.

Row sums of A297877.
The labeled version is A047864.
Equals A076278(n) + 1.
Cf. A005142 (connected).

A005142 = Cases[Import["https://oeis.org/A005142/b005142.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A005142] (* Jean-François Alcover, Mar 18 2020 *)

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Sep 05 2018

A117279 Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 3, 1, 10, 45, 110, 140, 60, 10, 1, 15, 105, 435, 1125, 1701, 1200, 480, 105, 10, 1, 21, 210, 1295, 5355, 14952, 26572, 26670, 17535, 7840, 2331, 420, 35, 1, 28, 378, 3220, 19075, 81228, 246414, 507424, 666015, 620900, 431368

Offset: 0

Vladeta Jovovic, Jun 23 2007

			Triangle begins:
  1;
  1;
  1,  1;
  1,  3,  3;
  1,  6, 15,  16,   3;
  1, 10, 45, 110, 140, 60, 10;
  ...

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Andrew Howroyd, Table of n, a(n) for n = 0..1403 (rows 0..25)

Row sums give A047864,
Columns k=1..5 are A000217(n-1), A050534, A053526, A053527, A053528.
The unlabeled version is A297877.
Cf. A052296, A033995.

nn=10;f[x_,y_]:=Sum[Sum[Binomial[n,k](1+y)^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Range[0,nn]!CoefficientList[Series[Exp[Log[f[x,y]]/2],{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Sep 05 2013 *)

T(n)={[Vecrev(p) | p<-Vec(serlaplace(sqrt(sum(k=0, n, exp(x*(1+y)^k + O(x*x^n))*x^k/k! ))))]}
{ my(A=T(6)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 10 2022

E.g.f.: sqrt(Sum_{n>=0} exp(x*(1+q)^n)*x^n/n!).

cp's OEIS Frontend

A033995 Number of bipartite graphs with n nodes.

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