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