A193589 - OEIS

%I #6 Mar 30 2012 18:57:38

%S 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,

%T 1065,2690,4575,1,14,102,515,2006,6276,15547,26398,1,16,133,766,3420,

%U 12468,37711,92124,156233,1,18,168,1085,5439,22412,78030,230277

%N Augmentation of the Fibonacci triangle A193588.  See Comments.

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

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

%C (col 3)=A033537.

%e First 5 rows of A193588:

%e 1

%e 1....2

%e 1....2....3

%e 1....2....3....5

%e 1....2....3....5....8

%e First 5 rows of A193589:

%e 1

%e 1....2

%e 1....4....7

%e 1....6....18...31

%e 1....8....33...90...154

%t p[n_, k_] := Fibonacci[k + 2]

%t Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A193588 *)

%t m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]

%t TableForm[m[4]]

%t w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];

%t v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};

%t v[n_] := v[n - 1].m[n]

%t TableForm[Table[v[n], {n, 0, 6}]]  (* A193589 *)

%t Flatten[Table[v[n], {n, 0, 8}]]

%Y Cf. A193091, A193588.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Jul 31 2011