A198261 Triangular array read by rows T(n,k) is the number of simple labeled graphs on n nodes with exactly k isolated nodes, 0<=k<=n.

1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 41, 16, 6, 0, 1, 768, 205, 40, 10, 0, 1, 27449, 4608, 615, 80, 15, 0, 1, 1887284, 192143, 16128, 1435, 140, 21, 0, 1, 252522481, 15098272, 768572, 43008, 2870, 224, 28, 0, 1

Offset: 0

Author

Geoffrey Critzer, Oct 22 2011

Keywords

• nonn
• tabl

Comments

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

First column (k=0) is A006129.

Examples

			Triangle begins:
  1;
  0,       1;
  1,       0,      1;
  4,       3,      0,     1;
  41,      16,     6,     0,    1;
  768,     205,    40,    10,   0,   1;
  27449,   4608,   615,   80,   15,  0,  1;
  1887284, 192143, 16128, 1435, 140, 21, 0, 1;
		

Crossrefs

Cf. A006125, A006129.

Programs

Mathematica

g=Sum[2^Binomial[n, 2]x^n/n!, {n, 0, 20}]; Transpose[Table[Range[0, 10]! CoefficientList[Series[(x^n/n!)( g/Exp[x]), {x, 0, 10}], x], {n, 0, 8}]]//Grid

Formula

E.g.f. for column k: x^k/k! *A(x)/exp(x) where A(x) is the e.g.f. for A006125.

T(n,n) = 1 (the empty graph). - Geoffrey Critzer, Nov 11 2011

T(n,n-1) = 0. - Geoffrey Critzer, Nov 11 2011