A193606 Augmentation of the triangle A193605. See Comments.

1, 1, 3, 1, 7, 17, 1, 12, 57, 127, 1, 18, 134, 531, 1125, 1, 25, 265, 1556, 5513, 11279, 1, 33, 470, 3793, 19152, 62675, 124837, 1, 42, 772, 8175, 55297, 250524, 771121, 1502679, 1, 52, 1197, 16087, 140269, 834879, 3478204, 10185019, 19480445

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..44.

Example

First five rows of A193606:
1
1...3
1...7....17
1...12...57....127
1...18...134...531...1125

Mathematica

u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}]
p[n_, k_] := Sum[u[n, h], {h, 0, k}]
Table[p[n, k], {n, 0, 12}, {k, 0, n}] (* A193695 *)
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}]] (* A193606 *)
Flatten[Table[v[n], {n, 0, 8}]]

Crossrefs

Cf. A193605, A193091.

Sequence in context: A232295 A228763 A101845 * A221328 A280610 A280416

Adjacent sequences: A193603 A193604 A193605 * A193607 A193608 A193609

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 31 2011

Status

approved