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%I #40 Apr 13 2015 09:42:50
%S 1,1,1,2,4,3,5,15,19,10,14,56,99,90,36,42,210,476,603,427,137,132,792,
%T 2190,3536,3507,2032,543,429,3003,9801,19185,24251,19800,9702,2219,
%U 1430,11440,43043,98890,151295,157716,109520,46472,9285,4862,43758
%N Triangle read by rows: T(n,k) = Sum_{i=n-k..n} C(k-1,n-i)*C(i,n-k)*C(2*i,i)/(i+1).
%F G.f.: (1-sqrt(1-4*(x/(1-x)+y)))/(2*(x/(1-x)+y)).
%e 1;
%e 1, 1;
%e 2, 4, 3;
%e 5, 15, 19, 10;
%e 14, 56, 99, 90, 36;
%e 42, 210, 476, 603, 427, 137;
%t T[n_, k_] := SeriesCoefficient[1-Sqrt[1-4*(x/(1-x)+y)]/(2*(x/(1-x)+y)), {x, 0, n}, {y, 0, k}]; Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Apr 13 2015 *)
%o (Maxima)
%o T(n,m):=sum((binomial(m-1,n-i)*binomial(i,n-m)*binomial(2*i,i))/(i+1),i,n-m,n);
%Y Cf. A000108 (first column), A002212 (right diagonal).
%K nonn,tabl
%O 0,4
%A _Vladimir Kruchinin_, Apr 10 2015
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