A193559 Augmentation of the triangular array |A123191|. See Comments.

1, 1, 1, 1, 4, 2, 1, 7, 17, 7, 1, 10, 41, 82, 32, 1, 13, 74, 238, 434, 166, 1, 16, 116, 502, 1412, 2446, 926, 1, 19, 167, 901, 3317, 8587, 14405, 5419, 1, 22, 227, 1462, 6581, 21802, 53381, 87610, 32816, 1, 25, 296, 2212, 11717, 46681, 143666, 338038

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.

Examples

			First five rows of |A123191|:
1
1...1
1...3...1
1...3...3...1
1...3...3...3...1
First 5 rows of A193559:
1
1...1
1...4...2
1...7...17...7
1...10..41...82...32
		

Crossrefs

Cf. A193091.

Programs

Mathematica

p[n_, k_] := If[Or[k == 0, k == n], 1, 3]
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* Abs. value of A123191 *)
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}]] (* A193559 *)
Flatten[Table[v[n], {n, 0, 10}]]