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A193606 Augmentation of the triangle A193605. See Comments.

%I A193606 #5 Mar 30 2012 18:57:38
%S A193606 1,1,3,1,7,17,1,12,57,127,1,18,134,531,1125,1,25,265,1556,5513,11279,
%T A193606 1,33,470,3793,19152,62675,124837,1,42,772,8175,55297,250524,771121,
%U A193606 1502679,1,52,1197,16087,140269,834879,3478204,10185019,19480445
%N A193606 Augmentation of the triangle A193605.  See Comments.
%C A193606 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%e A193606 First five rows of A193606:
%e A193606 1
%e A193606 1...3
%e A193606 1...7....17
%e A193606 1...12...57....127
%e A193606 1...18...134...531...1125
%t A193606 u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}]
%t A193606 p[n_, k_] := Sum[u[n, h], {h, 0, k}]
%t A193606 Table[p[n, k], {n, 0, 12}, {k, 0, n}] (* A193695 *)
%t A193606 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193606 TableForm[m[4]]
%t A193606 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193606 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193606 v[n_] := v[n - 1].m[n]
%t A193606 TableForm[Table[v[n], {n, 0, 6}]] (* A193606 *)
%t A193606 Flatten[Table[v[n], {n, 0, 8}]]
%Y A193606 Cf. A193605, A193091.
%K A193606 nonn,tabl
%O A193606 0,3
%A A193606 _Clark Kimberling_, Jul 31 2011