A228859 Triangular array read by rows. T(n,k) is the number of labeled bipartite graphs on n nodes having exactly k connected components; n>=1, 1<=k<=n.

1, 1, 1, 3, 3, 1, 19, 15, 6, 1, 195, 125, 45, 10, 1, 3031, 1545, 480, 105, 15, 1, 67263, 27307, 7035, 1400, 210, 21, 1, 2086099, 668367, 140098, 24045, 3430, 378, 28, 1, 89224635, 22427001, 3746925, 536214, 68355, 7434, 630, 36, 1

Offset: 1

Author

Geoffrey Critzer, Sep 05 2013

Keywords

• nonn
• tabl

Comments

The Bell transform of A001832(n+1) (without column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016

Examples

			1,
1, 1,
3, 3, 1,
19, 15, 6, 1,
195, 125, 45, 10, 1,
3031, 1545, 480, 105, 15, 1,
		

Crossrefs

Row sums are A047864.

Column 1 is A001832.

Cf. A047863.

Programs

Mathematica

nn=9;f[x_]:=Sum[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Drop[Range[0,nn]!CoefficientList[Series[Exp[y Log[f[x]]/2],{x,0,nn}],{x,y}],1]]//Grid

Sage

# uses[bell_matrix from A264428, A001832]
# Adds 1,0,0,0,... as column 0 to the triangle.
bell_matrix(lambda n: A001832(n+1), 8) # Peter Luschny, Jan 21 2016

Formula

E.g.f.: sqrt(A(x)^y) where A(x) is the e.g.f. for A047863.

Sum_{k=1..n} T(n,k)*2^k = A047863(n).