A228550 Triangular array read by rows: T(n,k) is the number of simple labeled graphs with n vertices and k components such that each vertex has even degree; n >= 1, 1 <= k <= n.

1, 0, 1, 1, 0, 1, 3, 4, 0, 1, 38, 15, 10, 0, 1, 720, 238, 45, 20, 0, 1, 26614, 5145, 868, 105, 35, 0, 1, 1858122, 215355, 21000, 2408, 210, 56, 0, 1, 250586792, 16797942, 980371, 64260, 5628, 378, 84, 0, 1, 66121926720, 2509697144, 84370230, 3306415, 163800, 11676, 630, 120, 0, 1

Offset: 1

Author

Geoffrey Critzer, Aug 27 2013

Keywords

• nonn
• tabl

Comments

The Bell transform of A033678(n+1). For the definition of the Bell transform, see A264428. - Peter Luschny, Jan 17 2016

Examples

			T(3,1) = 1 counts the complete graph on 3 labeled vertices.
T(3,3) = 1 counts the empty graph (no edges) on 3 labeled vertices.
Triangular array T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
    1;
    0,    1;
    1,    0,   1;
    3,    4,   0,   1;
   38,   15,  10,   0,  1;
  720,  238,  45,  20,  0, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..45, flattened

Crossrefs

Cf. A006125, A033678, A264428.

Programs

Mathematica

nn = 8; e = Sum[2^Binomial[n - 1, 2] x^n/n!, {n, 1, nn}];
  Prepend[Drop[Map[Insert[#, 0, -2] &,
    Map[Select[#, # > 0 &] &,
     Range[0, nn]! CoefficientList[
       Series[(e + 1)^y, {x, 0, nn}], {x, y}]]], 2], {1}] // Grid

Sage

# uses[bell_matrix from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: A033678(n+1), 9) # Peter Luschny, Jan 17 2016

Formula

E.g.f.: (A(x) + 1)^y, where A(x) = Sum_{n>=1} 2^C(n-1,2) * x^n/n!.

Row sums are 2^binomial(n-1,2) = A006125(n-1).

Column 1 is A033678 (because a connected graph has only one component).