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Augmentation of the Catalan triangle, A009766. See Comments.
3

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

%S 1,1,1,1,3,3,1,6,14,14,1,10,41,86,86,1,15,95,327,645,645,1,21,190,965,

%T 2991,5662,5662,1,28,343,2410,10684,30827,56632,56632,1,36,574,5334,

%U 31969,128959,352936,633545,633545,1,45,906,10766,83860,449435

%N Augmentation of the Catalan triangle, A009766. See Comments.

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

%C Regarding A193560, if the triangle is written as (w(n,k)), then w(n,n)=A127715(n).

%e First 5 rows of A193560:

%e 1

%e 1...1

%e 1...3...3

%e 1...6...14...14

%e 1...10..41...86...86

%t p[n_, k_] := ((n - k + 1)/(n + 1)) (n + k)!/(n!*k!) (* Catalan triangle, A009766 *)

%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}]] (* A193560 *)

%t Flatten[Table[v[n], {n, 0, 10}]]

%Y Cf. A193091.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Jul 30 2011