A193589 Augmentation of the Fibonacci triangle A193588. See Comments.

1, 1, 2, 1, 4, 7, 1, 6, 18, 31, 1, 8, 33, 90, 154, 1, 10, 52, 185, 481, 820, 1, 12, 75, 324, 1065, 2690, 4575, 1, 14, 102, 515, 2006, 6276, 15547, 26398, 1, 16, 133, 766, 3420, 12468, 37711, 92124, 156233, 1, 18, 168, 1085, 5439, 22412, 78030, 230277

Offset

0,3

Comments

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

Regarding A193589, if the triangle is written as (w(n,k)), then w(n,n)=A007863(n); w(n,n-1)=A011270; and

(col 3)=A033537.

Links

Table of n, a(n) for n=0..52.

Example

First 5 rows of A193588:
1
1....2
1....2....3
1....2....3....5
1....2....3....5....8
First 5 rows of A193589:
1
1....2
1....4....7
1....6....18...31
1....8....33...90...154

Mathematica

p[n_, k_] := Fibonacci[k + 2]
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A193588 *)
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}]]  (* A193589 *)
Flatten[Table[v[n], {n, 0, 8}]]

Crossrefs

Cf. A193091, A193588.

Sequence in context: A234946 A223092 A071948 * A187115 A121722 A193591

Adjacent sequences: A193586 A193587 A193588 * A193590 A193591 A193592

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 31 2011

Status

approved