%I #5 Mar 30 2012 18:57:38

%S 1,2,1,8,8,3,64,88,62,19,1024,1664,1568,896,233,32768,58368,64128,

%T 49248,23890,5385,2097152,3932160,4703232,4249728,2800488,1179656,

%U 233787,268435456,517996544,649887744,645547008,507802304,293645688,108978862

%N Augmentation of the triangle A055248. 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 A193604:

%e 1

%e 2.....1

%e 8.....8.....3

%e 64....88....62....19

%e 1024..1664..1568..896...233

%t u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}] (* A055248 *)

%t p[n_, k_] := u[n, n - k]

%t Table[p[n, k], {n, 0, 5}, {k, 0, n}]

%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}]] (* A193604 *)

%t Flatten[Table[v[n], {n, 0, 8}]]

%Y Cf. A055248, A193091.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Jul 31 2011