A193607 Augmentation of the triangle A011973. See Comments.

1, 1, 1, 1, 4, 2, 1, 9, 20, 7, 1, 16, 80, 131, 37, 1, 25, 220, 806, 1085, 265, 1, 36, 490, 3130, 9360, 10952, 2402, 1, 49, 952, 9325, 48224, 124498, 130852, 26371, 1, 64, 1680, 23317, 183569, 813886, 1876101, 1809430, 340272

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.

(col 2): A000290 (the squares)

(col 3)= 2*A002419

Examples

			First five rows of A193607:
1
1...1
1...4....2
1...9....20...7
1...16...80...131...37
		

Crossrefs

Cf. A011973, A193091, A002419.

Programs

Mathematica

p[n_, k_] := Binomial[2 n - k, k];
Table[p[n, k], {n, 0, 7}, {k, 0, n}]  (* A011973 *)
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, 9}]] (* A193607 *)
Flatten[Table[v[n], {n, 0, 8}]]