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%I #10 Apr 30 2015 21:03:43
%S 1,3,1,8,6,1,22,25,9,1,64,92,51,12,1,196,324,237,86,15,1,625,1128,996,
%T 484,130,18,1,2055,3934,3966,2377,860,183,21,1,6917,13812,15335,10744,
%U 4845,1392,245,24,1,23713,48884,58359,46068,24603,8859,2107,316,27,1
%N Triangle, read by rows, T(n,k) = k*Sum_{i=0..n-k} C(2*i+2*k,i)*C(n-i-1,k-1)/(i+k) for 1 <= k <= n.
%F G.f.: 1/(1-(C(x)-1)/(1-x)*y)-1, where C(x) is g.f. of Catalan numbers (A000108).
%F T(n,n-1) = 3*(n-1) for n > 1. - _Derek Orr_, Apr 27 2015
%F T(n,n-2) = A062728(n-2) for n > 2. - _Derek Orr_, Apr 27 2015
%F T(n,1) = A014138(n). - _Derek Orr_, Apr 27 2015
%e Triangle starts:
%e 1;
%e 3, 1;
%e 8, 6, 1;
%e 22, 25, 9, 1;
%e 64, 92, 51, 12, 1;
%t Flatten@ Table[k Sum[Binomial[2 i + 2 k, i] Binomial[n - i - 1, k - 1]/(i + k), {i, 0, n - k}], {n, 10}, {k, n}] (* _Michael De Vlieger_, Apr 27 2015 *)
%o (Maxima)
%o T(n,k):=k*sum((binomial(2*i+2*k,i)*binomial(n-i-1,k-1))/(i+k),i,0,n-k);
%o (PARI) T(n,k)=k*sum(i=0,n-k,(binomial(2*i+2*k,i)*binomial(n-i-1,k-1))/(i+k))
%o for(n=1,10,for(k=1,n,print1(T(n,k),", "))) \\ _Derek Orr_, Apr 27 2015
%Y Cf. A014138.
%K nonn,tabl,easy
%O 1,2
%A _Vladimir Kruchinin_, Apr 26 2015
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