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A193560 Augmentation of the Catalan triangle, A009766. See Comments.

%I A193560 #5 Mar 30 2012 18:57:38
%S A193560 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 A193560 2991,5662,5662,1,28,343,2410,10684,30827,56632,56632,1,36,574,5334,
%U A193560 31969,128959,352936,633545,633545,1,45,906,10766,83860,449435
%N A193560 Augmentation of the Catalan triangle, A009766.  See Comments.
%C A193560 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193560 Regarding A193560, if the triangle is written as (w(n,k)), then w(n,n)=A127715(n).
%e A193560 First 5 rows of A193560:
%e A193560 1
%e A193560 1...1
%e A193560 1...3...3
%e A193560 1...6...14...14
%e A193560 1...10..41...86...86
%t A193560 p[n_, k_] := ((n - k + 1)/(n + 1)) (n + k)!/(n!*k!)  (* Catalan triangle, A009766 *)
%t A193560 Table[p[n, k], {n, 0, 5}, {k, 0, n}]
%t A193560 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193560 TableForm[m[4]]
%t A193560 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193560 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193560 v[n_] := v[n - 1].m[n]
%t A193560 TableForm[Table[v[n], {n, 0, 6}]]  (* A193560 *)
%t A193560 Flatten[Table[v[n], {n, 0, 10}]]
%Y A193560 Cf. A193091.
%K A193560 nonn,tabl
%O A193560 0,5
%A A193560 _Clark Kimberling_, Jul 30 2011