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A193631 Augmentation of the triangle given by p(n,k)=(3+(-1)^k)/2 for 0<=k<=n. See Comments.

%I A193631 #5 Mar 30 2012 18:57:38
%S A193631 1,2,1,4,4,5,8,12,22,17,16,32,72,88,89,32,80,208,328,474,417,64,192,
%T A193631 560,1056,1836,2364,2253,128,448,1440,3120,6168,9684,13038,11937,256,
%U A193631 1024,3584,8704,19040,34240,54800,71152,66737,512,2304,8704,23296
%N A193631 Augmentation of the triangle given by p(n,k)=(3+(-1)^k)/2 for 0<=k<=n.  See Comments.
%C A193631 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193631 (column 2 of A193631)=A001787.
%e A193631 First five rows of the triangle P=p(n,k):
%e A193631 2
%e A193631 2...1
%e A193631 2...1...2
%e A193631 2...1...2...1
%e A193631 2...1...2...1...2
%e A193631 First five rows of A193631:
%e A193631 1
%e A193631 2....1
%e A193631 4....4....5
%e A193631 8....12...22...17
%e A193631 16...32...72...88...89
%t A193631 p[n_, k_] := (3 + (-1)^k)/2;
%t A193631 Table[p[n, k], {n, 0, 7}, {k, 0, n}]
%t A193631 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193631 TableForm[m[4]]
%t A193631 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193631 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193631 v[n_] := v[n - 1].m[n]
%t A193631 TableForm[Table[v[n], {n, 0, 6}]] (* A193631 *)
%t A193631 Flatten[Table[v[n], {n, 0, 8}]]
%Y A193631 Cf. A193091, A001787.
%K A193631 nonn,tabl
%O A193631 0,2
%A A193631 _Clark Kimberling_, Aug 01 2011