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

%S 1,2,1,4,4,5,8,12,22,17,16,32,72,88,89,32,80,208,328,474,417,64,192,

%T 560,1056,1836,2364,2253,128,448,1440,3120,6168,9684,13038,11937,256,

%U 1024,3584,8704,19040,34240,54800,71152,66737,512,2304,8704,23296

%N Augmentation of the triangle given by p(n,k)=(3+(-1)^k)/2 for 0<=k<=n. See Comments.

%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

%C (column 2 of A193631)=A001787.

%e First five rows of the triangle P=p(n,k):

%e 2

%e 2...1

%e 2...1...2

%e 2...1...2...1

%e 2...1...2...1...2

%e First five rows of A193631:

%e 1

%e 2....1

%e 4....4....5

%e 8....12...22...17

%e 16...32...72...88...89

%t p[n_, k_] := (3 + (-1)^k)/2;

%t Table[p[n, k], {n, 0, 7}, {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}]] (* A193631 *)

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

%Y Cf. A193091, A001787.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 01 2011