A193597 Augmentation of the triangle A193596. See Comments.

1, 1, 1, 1, 2, 2, 1, 4, 6, 5, 1, 6, 17, 21, 16, 1, 9, 34, 78, 86, 61, 1, 12, 69, 201, 397, 401, 269, 1, 16, 116, 522, 1282, 2250, 2113, 1350, 1, 20, 194, 1074, 4099, 8900, 14187, 12509, 7650, 1, 25, 292, 2172, 10078, 34044, 67316, 99102, 82713, 48634, 1

Offset: 0

Author

Clark Kimberling, Jul 31 2011

Keywords

• nonn
• tabl

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Examples

			First 5 rows of A193596:
1
1...1
1...1...1
1...2...2...1
1...2...3...2...1
		

Crossrefs

Cf. A193596, A193091.

Programs

Mathematica

p[n_, k_] := Ceiling[Binomial[n, k]/2]
Table[p[n, k], {n, 0, 10}, {k, 0, n}]
Flatten[%]  (* A193596 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 10}]] (* A193597 *)
Flatten[Table[v[n], {n, 0, 10}]]