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%I #11 Aug 29 2017 16:02:35
%S 1,1,1,2,2,1,5,5,3,1,14,14,10,4,1,42,42,35,17,5,1,132,132,126,74,26,6,
%T 1,429,429,462,326,137,37,7,1,1430,1430,1716,1446,726,230,50,8,1,4862,
%U 4862,6435,6441,3858,1434,359,65,9,1,16796,16796,24310,28770,20532,8952,2582,530,82,10,1
%N A number triangle based on the Catalan numbers.
%C Columns include A000108, A001700, A049027(n+1), A076025(n+1). Rows sums are A110489, diagonal sums are A110490.
%H G. C. Greubel, <a href="/A110488/b110488.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F T(n, k) = Sum_{j=0..(n-k)} 2*(j+1)*(k-1)^j*C(2(n-k)+1, n-k-j)/(n-k+j+2)}.
%F Column k has g.f. x^k*c(x)/(1-k*x*c(x)) where c(x) is the g.f. of A000108.
%F T(n,0) = Catalan(n), T(n,1) = Catalan(n), T(n,n) = 1. - _G. C. Greubel_, Aug 28 2017
%e Rows begin
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 5, 5, 3, 1;
%e 14, 14, 10, 4, 1;
%e 42, 42, 35, 17, 5, 1;
%t T[n_, 0] := CatalanNumber[n]; T[n_, 1] := CatalanNumber[n]; T[n_, n_] := 1; T[n_, k_] := Sum[2*(j + 1)*(k - 1)^j*Binomial[2 (n - k) + 1, n - k - j]/(n - k + j + 2), {j, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Aug 28 2017 *)
%K easy,nonn,tabl
%O 0,4
%A _Paul Barry_, Jul 22 2005
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