A193631 Augmentation of the triangle given by p(n,k)=(3+(-1)^k)/2 for 0<=k<=n. See Comments.

1, 2, 1, 4, 4, 5, 8, 12, 22, 17, 16, 32, 72, 88, 89, 32, 80, 208, 328, 474, 417, 64, 192, 560, 1056, 1836, 2364, 2253, 128, 448, 1440, 3120, 6168, 9684, 13038, 11937, 256, 1024, 3584, 8704, 19040, 34240, 54800, 71152, 66737, 512, 2304, 8704, 23296

Offset: 0

Author

Clark Kimberling, Aug 01 2011

Keywords

• nonn
• tabl

Comments

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

(column 2 of A193631)=A001787.

Examples

			First five rows of the triangle P=p(n,k):
2
2...1
2...1...2
2...1...2...1
2...1...2...1...2
First five rows of A193631:
1
2....1
4....4....5
8....12...22...17
16...32...72...88...89
		

Crossrefs

Cf. A193091, A001787.

Programs

Mathematica

p[n_, k_] := (3 + (-1)^k)/2;
Table[p[n, k], {n, 0, 7}, {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}]] (* A193631 *)
Flatten[Table[v[n], {n, 0, 8}]]