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

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

Author

Vladeta Jovovic, Jun 23 2007

Keywords

• nonn
• tabf

Examples

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

References

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

Links

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

Crossrefs

Row sums give A047864,

Columns k=1..5 are A000217(n-1), A050534, A053526, A053527, A053528.

The unlabeled version is A297877.

Cf. A052296, A033995.

Programs

Mathematica

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 *)

PARI

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

Formula

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