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%I #7 Mar 30 2012 18:57:38
%S 1,1,2,1,5,7,1,9,28,33,1,14,74,181,191,1,20,159,637,1333,1297,1,27,
%T 300,1767,5906,11029,10063,1,35,517,4190,20256,59324,101351,87669,1,
%U 44,833,8873,58339,244125,645146,1024949,847015,1,54,1274,17241,147680
%N Augmentation of the triangle A074909. See Comments.
%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C Regarding A193630, writing the general term as w(n,k),
%C w(n,n): A104981
%C w(n,n-1): A156629
%e First five rows of A193607:
%e 1
%e 1...2
%e 1...5....7
%e 1...9....28...33
%e 1...14...74...181...191
%t p[n_, k_] := Binomial[n + 1, k];
%t Table[p[n, k], {n, 0, 7}, {k, 0, n}] (* A074909 *)
%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}]] (* A193630 *)
%t Flatten[Table[v[n], {n, 0, 8}]]
%Y Cf. A193091, A074909.
%K nonn,tabl
%O 0,3
%A _Clark Kimberling_, Aug 01 2011
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