login
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).
0

%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