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.

1, 1, 1, 5, 10, 5, 28, 84, 84, 28, 165, 660, 990, 660, 165, 1001, 5005, 10010, 10010, 5005, 1001, 6188, 37128, 92820, 123760, 92820, 37128, 6188, 38760, 271320, 813960, 1356600, 1356600, 813960, 271320, 38760, 245157, 1961256, 6864396, 13728792, 17160990, 13728792, 6864396, 1961256, 245157

Offset: 0

Author

Vladimir Kruchinin, Feb 11 2023

Keywords

• nonn
• tabl

Examples

			Triangle begins:
     1;
     1,    1;
     5,   10,     5;
    28,   84,    84,    28;
   165,  660,   990,   660,  165;
  1001, 5005, 10010, 10010, 5005, 1001;
		

Crossrefs

Cf. A025174, A090763, A360546.

Programs

Maple

T := (n, k) -> ifelse(n = 0, 1, binomial(n, k)*binomial(3*n - 1, n)/2):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;

Maxima

T(n,m):=1/2*binomial(n+1,m)*binomial(3*n+2,n+1);

Formula

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))).