%I #11 Feb 15 2023 15:24:07
%S 1,3,3,10,20,10,35,105,105,35,126,504,756,504,126,462,2310,4620,4620,
%T 2310,462,1716,10296,25740,34320,25740,10296,1716,6435,45045,135135,
%U 225225,225225,135135,45045,6435,24310,194480,680680,1361360,1701700,1361360,680680,194480,24310
%N Triangle T(n, m) = (n - m + 1)*C(2*n + 1, m)*C(2*n - m + 2, n - m + 1)/(2*n - m + 2).
%F G.f.: 2/(1 - 4*x + sqrt(1 - 4*x - 4*x*y) - 4*x*y).
%F T(n, k) = binomial(n, k)*CatalanNumber(n)*(2*n + 1). - _Peter Luschny_, Feb 15 2023
%e Triangle T(n, m) starts:
%e [0] 1;
%e [1] 3, 3;
%e [2] 10, 20, 10;
%e [3] 35, 105, 105, 35;
%e [4] 126, 504, 756, 504, 126;
%e [5] 462, 2310, 4620, 4620, 2310, 462;
%e [6] 1716, 10296, 25740, 34320, 25740, 10296, 1716;
%e [7] 6435, 45045, 135135, 225225, 225225, 135135, 45045, 6435;
%p CatalanNumber := n -> binomial(2*n, n)/(n + 1):
%p T := (n, k) -> (2*n + 1)*CatalanNumber(n)*binomial(n, k):
%p seq(seq(T(n, k), k = 0..n), n = 0..8); # _Peter Luschny_, Feb 15 2023
%o (Maxima)
%o T(n,m):=if n<m then 0 else ((n-m+1)*binomial(2*n+1,m)*binomial(2*n-m+2,n-m+1))/(2*n-m+2);
%Y Cf. A001700, A085880, A069720 (row sums).
%K nonn,tabl
%O 0,2
%A _Vladimir Kruchinin_, Feb 15 2023