A285529 Triangle read by rows: T(n,k) is the number of nodes of degree k counted over all simple labeled graphs on n nodes, n>=1, 0<=k<=n-1.

1, 2, 2, 6, 12, 6, 32, 96, 96, 32, 320, 1280, 1920, 1280, 320, 6144, 30720, 61440, 61440, 30720, 6144, 229376, 1376256, 3440640, 4587520, 3440640, 1376256, 229376, 16777216, 117440512, 352321536, 587202560, 587202560, 352321536, 117440512, 16777216

Offset: 1

Author

Geoffrey Critzer, Apr 20 2017

Keywords

• nonn
• tabl

Examples

			1,
2,   2,
6,   12,   6,
32,  96,   96,   32,
320, 1280, 1920, 1280, 320,
...
		

Crossrefs

Row sums give A095340.

Columns for k=0-3: A123903, A095338, A038094, A038096.

Programs

Mathematica

nn = 9; Map[Select[#, # > 0 &] &,
  Drop[Transpose[Table[A[z_] := Sum[Binomial[n, k] 2^Binomial[n, 2] z^n/n!, {n, 0, nn}];Range[0, nn]! CoefficientList[Series[z A[z], {z, 0, nn}], z], {k,0, nn - 1}]], 1]] // Grid

Formula

E.g.f. for column k: x * Sum_{n>=0} binomial(n,k)*2^binomial(n,2)*x^n/n!.

Sum_{k=1..n-1} T(n,k)*k/2 = A095351(n).

T(n,k) = n*binomial(n-1,k)*2^binomial(n-1,2). - Alois P. Heinz, Apr 21 2017