A193603 Augmentation of the triangle A008949. See Comments.

1, 1, 2, 1, 5, 8, 1, 9, 30, 44, 1, 14, 77, 212, 296, 1, 20, 163, 700, 1712, 2312, 1, 27, 305, 1877, 6882, 15476, 20384, 1, 35, 523, 4365, 22380, 73240, 154424, 199376, 1, 44, 840, 9134, 62479, 280630, 841312, 1683992, 2138336

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. The right edge of the triangle A193603 is A111537.

Examples

			First five rows of A193603:
1
1...2
1...5....8
1...9...30....44
1...14...77..212...296
		

Crossrefs

Cf. A008949, A193091, A111537.

Programs

Mathematica

p[n_, k_] := Sum[Binomial[n, h], {h, 0, k}] (* A008949 *)
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}]] (* A193603 *)
Flatten[Table[v[n], {n, 0, 8}]]