A336667 Triangular array read by rows. T(n,k) is the number of closed walks of length 2n along the edges of a cube based at vertex v that return to v exactly k times, n>=0, 0<=k<=n.

1, 0, 3, 0, 12, 9, 0, 84, 72, 27, 0, 588, 648, 324, 81, 0, 4116, 5544, 3564, 1296, 243, 0, 28812, 45864, 35748, 16848, 4860, 729, 0, 201684, 370440, 337932, 193104, 72900, 17496, 2187

Offset: 0

Author

Geoffrey Critzer, Jul 29 2020

Keywords

• nonn
• tabl
• easy

Examples

			Triangle T(n,k) begins:
  1;
  0,   3;
  0,  12,   9;
  0,  84,  72,  27;
  0, 588, 648, 324, 81;
  ...
		

Links

• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 340.

Crossrefs

Cf. A054879 (row sums), A328778 (column k=1).

Programs

Mathematica

Table[nn = n; CoefficientList[Series[(1 - 7 z^2)/(1 - (7 + 3 u) z^2 + 9 u z^4), {z, 0, nn}], {z,u}][[-1]], {n, 0, 15, 2}] // Grid

Formula

O.g.f.: (1 - 7*x^2)/(1 - 7*x^2 - 3*y*x^2 + 9*y*x^4).