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%I #5 Mar 30 2012 18:57:38
%S 1,1,1,1,2,2,1,4,6,5,1,6,17,21,16,1,9,34,78,86,61,1,12,69,201,397,401,
%T 269,1,16,116,522,1282,2250,2113,1350,1,20,194,1074,4099,8900,14187,
%U 12509,7650,1,25,292,2172,10078,34044,67316,99102,82713,48634,1
%N Augmentation of the triangle A193596. See Comments.
%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%e First 5 rows of A193596:
%e 1
%e 1...1
%e 1...1...1
%e 1...2...2...1
%e 1...2...3...2...1
%t p[n_, k_] := Ceiling[Binomial[n, k]/2]
%t Table[p[n, k], {n, 0, 10}, {k, 0, n}]
%t Flatten[%] (* A193596 *)
%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, 10}]] (* A193597 *)
%t Flatten[Table[v[n], {n, 0, 10}]]
%Y Cf. A193596, A193091.
%K nonn,tabl
%O 0,5
%A _Clark Kimberling_, Jul 31 2011
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