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A193603 Augmentation of the triangle A008949. See Comments.

%I A193603 #6 Mar 30 2012 18:57:38
%S A193603 1,1,2,1,5,8,1,9,30,44,1,14,77,212,296,1,20,163,700,1712,2312,1,27,
%T A193603 305,1877,6882,15476,20384,1,35,523,4365,22380,73240,154424,199376,1,
%U A193603 44,840,9134,62479,280630,841312,1683992,2138336
%N A193603 Augmentation of the triangle A008949.  See Comments.
%C A193603 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.  The right edge of the triangle A193603 is A111537.
%e A193603 First five rows of A193603:
%e A193603 1
%e A193603 1...2
%e A193603 1...5....8
%e A193603 1...9...30....44
%e A193603 1...14...77..212...296
%t A193603 p[n_, k_] := Sum[Binomial[n, h], {h, 0, k}] (* A008949 *)
%t A193603 Table[p[n, k], {n, 0, 5}, {k, 0, n}]
%t A193603 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193603 TableForm[m[4]]
%t A193603 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193603 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193603 v[n_] := v[n - 1].m[n]
%t A193603 TableForm[Table[v[n], {n, 0, 6}]] (* A193603 *)
%t A193603 Flatten[Table[v[n], {n, 0, 8}]]
%Y A193603 Cf. A008949, A193091, A111537.
%K A193603 nonn,tabl
%O A193603 0,3
%A A193603 _Clark Kimberling_, Jul 31 2011