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A193589
Augmentation of the Fibonacci triangle A193588. See Comments.
4
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.
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
Sequence in context: A234946 A223092 A071948 * A187115 A121722 A193591
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jul 31 2011
STATUS
approved