A122881 Triangle read by rows: number of Catalan paths of 2n steps of all values less than or equal to m.

1, 1, 2, 1, 2, 5, 1, 2, 5, 13, 1, 2, 5, 14, 34, 1, 2, 5, 14, 42, 89, 1, 2, 5, 14, 42, 131, 233, 1, 2, 5, 14, 42, 132, 417, 610, 1, 2, 5, 14, 42, 132, 429, 1341, 1597, 1, 2, 5, 14, 42, 132, 429, 1429, 4334, 4181

Offset: 1

Author

Gary W. Adamson, Sep 16 2006

Keywords

• nonn
• tabl

Comments

Convergents of k-th diagonals relate to (2k+3)-polygons; e.g., right border relates to the pentagon (N=5), next border relates to the heptagon (N=7). Convergents of the diagonals are 2 + 2*cos(2*Pi/N) and are roots to Morgan-Voyce polynomials. k2 diagonal = A080937, number of Catalan paths of 2n steps of all values less than or equal to 5. k3 diagonal = A080938, number of Catalan paths of 2n steps of all values less than or equal to 7.

Examples

			For the right border, odd-indexed Fibonacci numbers (1, 2, 5, 13, 34...), we begin with (M2) = [1, 1; 1, 0], then P2 = [1, -1; -1, 2] = 1/(M2)^2. Performing (P2)^n * [1,0] we extract the left vector (1, 2, 5, 13, ...), making it the right border of the triangle, k1 diagonal.
For the next diagonal going to the left, we begin with the Heptagonal matrix M3 = [1, 1, 1; 1, 1, 0; 1, 0, 0], take the inverse square (P3) and then perform the analogous operation getting 1, 2, 5, 14, 42, ...
First few rows of the triangle are:
  1;
  1, 2;
  1, 2, 5;
  1, 2, 5, 13;
  1, 2, 5, 14, 34;
  1, 2, 5, 14, 42, 89;
  1, 2, 5, 14, 42, 131, 233;
  1, 2, 5, 14, 42, 132, 417, 610;
  ...
		

Crossrefs

Cf. A112880, A001519, A000108, A080937, A080938.

Formula

Begin with polygonal matrices of the form (exemplified by the Heptagonal matrix M3: [1, 1, 1; 1, 1, 0; 1, 0, 0]). Let matrix P3 = 1 / M3^2; then for n X n matrices P2, P3, P4...perform P^n * [1, 0, 0] letting this vector = k-th diagonal of the triangle.