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A193559 Augmentation of the triangular array |A123191|. See Comments.

%I A193559 #5 Mar 30 2012 18:57:38
%S A193559 1,1,1,1,4,2,1,7,17,7,1,10,41,82,32,1,13,74,238,434,166,1,16,116,502,
%T A193559 1412,2446,926,1,19,167,901,3317,8587,14405,5419,1,22,227,1462,6581,
%U A193559 21802,53381,87610,32816,1,25,296,2212,11717,46681,143666,338038
%N A193559 Augmentation of the triangular array |A123191|.  See Comments.
%C A193559 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%e A193559 First five rows of |A123191|:
%e A193559 1
%e A193559 1...1
%e A193559 1...3...1
%e A193559 1...3...3...1
%e A193559 1...3...3...3...1
%e A193559 First 5 rows of A193559:
%e A193559 1
%e A193559 1...1
%e A193559 1...4...2
%e A193559 1...7...17...7
%e A193559 1...10..41...82...32
%t A193559 p[n_, k_] := If[Or[k == 0, k == n], 1, 3]
%t A193559 Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* Abs. value of A123191 *)
%t A193559 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193559 TableForm[m[4]]
%t A193559 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193559 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193559 v[n_] := v[n - 1].m[n]
%t A193559 TableForm[Table[v[n], {n, 0, 6}]] (* A193559 *)
%t A193559 Flatten[Table[v[n], {n, 0, 10}]]
%Y A193559 Cf. A193091.
%K A193559 nonn,tabl
%O A193559 0,5
%A A193559 _Clark Kimberling_, Jul 30 2011