A193630 Augmentation of the triangle A074909. See Comments.

1, 1, 2, 1, 5, 7, 1, 9, 28, 33, 1, 14, 74, 181, 191, 1, 20, 159, 637, 1333, 1297, 1, 27, 300, 1767, 5906, 11029, 10063, 1, 35, 517, 4190, 20256, 59324, 101351, 87669, 1, 44, 833, 8873, 58339, 244125, 645146, 1024949, 847015, 1, 54, 1274, 17241, 147680

Offset

0,3

Comments

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

Regarding A193630, writing the general term as w(n,k),

w(n,n): A104981

w(n,n-1): A156629

Links

Table of n, a(n) for n=0..49.

Example

First five rows of A193607:
1
1...2
1...5....7
1...9....28...33
1...14...74...181...191

Mathematica

p[n_, k_] := Binomial[n + 1, k];
Table[p[n, k], {n, 0, 7}, {k, 0, n}]  (* A074909 *)
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}]] (* A193630 *)
Flatten[Table[v[n], {n, 0, 8}]]

Crossrefs

Cf. A193091, A074909.

Sequence in context: A143892 A129321 A064642 * A144505 A369527 A370382

Adjacent sequences: A193627 A193628 A193629 * A193631 A193632 A193633

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 01 2011

Status

approved