A228315 Triangular array read by rows: T(n,k) is the number of rooted labeled simple graphs on {1,2,...,n} such that the root is in a component of size k; n>=1, 1<=k<=n.

1, 2, 2, 6, 6, 12, 32, 24, 48, 152, 320, 160, 240, 760, 3640, 6144, 1920, 1920, 4560, 21840, 160224, 229376, 43008, 26880, 42560, 152880, 1121568, 13063792, 16777216, 1835008, 688128, 680960, 1630720, 8972544, 104510336, 2012388736

Offset: 1

Author

Geoffrey Critzer, Aug 26 2013

Keywords

• nonn
• tabl

Comments

Row sums = A095340.

Column 1 = A123903.

T(n,k) = A223894(n,k)*k.

Diagonal = A053549.

Examples

			1;
2,    2;
6,    6,    12;
32,   24,   48,    152;
320,  160,  240,   760,    3640;
		

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7.

Links

• Alois P. Heinz, Rows n = 1..45, flattened
• Index entries for sequences related to graphs

Crossrefs

Cf. A070166.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
    end:
T:= (n, k)-> binomial(n, k)*k*b(k)*2^((n-k)*(n-k-1)/2):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 26 2013

Mathematica

nn = 10; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; a =
Drop[Range[0, nn]! CoefficientList[Series[Log[g], {x, 0, nn}], x],
  1]; Table[
  Table[Binomial[n, k] k a[[k]] 2^Binomial[n - k, 2], {k, 1, n}], {n,
   1, 7}] // Grid

Formula

T(n,k) = binomial(n,k)*k*A001187(k)*A006125(n-k).