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A360560 Triangle read by rows. T(n, k) = (1/2) * C(n, k) * C(3*n - 1, n) for n > 0 and T(0, 0) = 1.

%I A360560 #34 Feb 14 2023 08:56:57
%S A360560 1,1,1,5,10,5,28,84,84,28,165,660,990,660,165,1001,5005,10010,10010,
%T A360560 5005,1001,6188,37128,92820,123760,92820,37128,6188,38760,271320,
%U A360560 813960,1356600,1356600,813960,271320,38760,245157,1961256,6864396,13728792,17160990,13728792,6864396,1961256,245157
%N A360560 Triangle read by rows. T(n, k) = (1/2) * C(n, k) * C(3*n - 1, n) for n > 0 and T(0, 0) = 1.
%F A360560 G.f.: 1/2 + x*sqrt(3 + 3*y)*cot(arcsin((3*sqrt(3*x*(y + 1)))/2)/3)/ (2*sqrt(4*x - 27*x^2*(y + 1))).
%e A360560 Triangle begins:
%e A360560      1;
%e A360560      1,    1;
%e A360560      5,   10,     5;
%e A360560     28,   84,    84,    28;
%e A360560    165,  660,   990,   660,  165;
%e A360560   1001, 5005, 10010, 10010, 5005, 1001;
%p A360560 T := (n, k) -> ifelse(n = 0, 1, binomial(n, k)*binomial(3*n - 1, n)/2):
%p A360560 for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
%o A360560 (Maxima)
%o A360560 T(n,m):=1/2*binomial(n+1,m)*binomial(3*n+2,n+1);
%Y A360560 Cf. A025174, A090763, A360546.
%K A360560 nonn,tabl
%O A360560 0,4
%A A360560 _Vladimir Kruchinin_, Feb 11 2023