A189898 Triangular array read by rows. T(n,k) is the number of digraphs with n labeled nodes having exactly k undirected (or weak) components, n >= 1, 1 <= k <= n.

1, 3, 1, 54, 9, 1, 3834, 243, 18, 1, 1027080, 20790, 675, 30, 1, 1067308488, 6364170, 67635, 1485, 45, 1, 4390480193904, 7543111716, 23031540, 171045, 2835, 63, 1, 72022346388181584, 35217115838604, 30469951764, 63580545, 370440, 4914, 84, 1

Offset: 1

Author

Geoffrey Critzer, May 01 2011

Keywords

• nonn
• tabl

Comments

The Bell transform of A003027(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Examples

			1
3       1
54      9     1
3834    243   18   1
1027080 20790 675  30  1
		

Links

• Alois P. Heinz, Rows n = 1..32, flattened

Crossrefs

Column 1 = A003027, row sums = A053763, lower diagonal = A045943.

Programs

Maple

T:= (n, k)-> coeff(series(log(add(2^(i^2-i) *x^i/i!, i=0..n))^k /k!,
                   x, n+1), x, n) *n!:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, May 01 2011

Mathematica

a= Sum[4^Binomial[n,2]x^n/n!,{n,0,10}];
Transpose[Map[Drop[#, 1] &,Table[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}], x], {n, 1, 10}]]] // Grid

Sage

# uses[bell_matrix from A264428, A003027]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A003027(n+1), 10) # Peter Luschny, Jan 18 2016

Formula

E.g.f. for column k: log(A(x))^k/k! where A(x) is the e.g.f. for A053763.

Extensions

Name clarified by Andrew Howroyd, Jan 11 2022