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A193589 Augmentation of the Fibonacci triangle A193588. See Comments.

%I A193589 #6 Mar 30 2012 18:57:38
%S A193589 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 A193589 1065,2690,4575,1,14,102,515,2006,6276,15547,26398,1,16,133,766,3420,
%U A193589 12468,37711,92124,156233,1,18,168,1085,5439,22412,78030,230277
%N A193589 Augmentation of the Fibonacci triangle A193588.  See Comments.
%C A193589 For an introduction to the unary operation augmentation as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193589 Regarding A193589, if the triangle is written as (w(n,k)), then w(n,n)=A007863(n); w(n,n-1)=A011270; and
%C A193589 (col 3)=A033537.
%e A193589 First 5 rows of A193588:
%e A193589 1
%e A193589 1....2
%e A193589 1....2....3
%e A193589 1....2....3....5
%e A193589 1....2....3....5....8
%e A193589 First 5 rows of A193589:
%e A193589 1
%e A193589 1....2
%e A193589 1....4....7
%e A193589 1....6....18...31
%e A193589 1....8....33...90...154
%t A193589 p[n_, k_] := Fibonacci[k + 2]
%t A193589 Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A193588 *)
%t A193589 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193589 TableForm[m[4]]
%t A193589 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193589 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193589 v[n_] := v[n - 1].m[n]
%t A193589 TableForm[Table[v[n], {n, 0, 6}]]  (* A193589 *)
%t A193589 Flatten[Table[v[n], {n, 0, 8}]]
%Y A193589 Cf. A193091, A193588.
%K A193589 nonn,tabl
%O A193589 0,3
%A A193589 _Clark Kimberling_, Jul 31 2011