A193560 Augmentation of the Catalan triangle, A009766. See Comments.

1, 1, 1, 1, 3, 3, 1, 6, 14, 14, 1, 10, 41, 86, 86, 1, 15, 95, 327, 645, 645, 1, 21, 190, 965, 2991, 5662, 5662, 1, 28, 343, 2410, 10684, 30827, 56632, 56632, 1, 36, 574, 5334, 31969, 128959, 352936, 633545, 633545, 1, 45, 906, 10766, 83860, 449435

Offset: 0

Author

Clark Kimberling, Jul 30 2011

Keywords

• nonn
• tabl

Comments

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

Regarding A193560, if the triangle is written as (w(n,k)), then w(n,n)=A127715(n).

Examples

			First 5 rows of A193560:
1
1...1
1...3...3
1...6...14...14
1...10..41...86...86
		

Crossrefs

Cf. A193091.

Programs

Mathematica

p[n_, k_] := ((n - k + 1)/(n + 1)) (n + k)!/(n!*k!)  (* Catalan triangle, A009766 *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
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, 6}]]  (* A193560 *)
Flatten[Table[v[n], {n, 0, 10}]]