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A360651 Triangle T(n, m) = (n - m + 1)*C(2*n + 1, m)*C(2*n - m + 2, n - m + 1)/(2*n - m + 2).

%I A360651 #11 Feb 15 2023 15:24:07
%S A360651 1,3,3,10,20,10,35,105,105,35,126,504,756,504,126,462,2310,4620,4620,
%T A360651 2310,462,1716,10296,25740,34320,25740,10296,1716,6435,45045,135135,
%U A360651 225225,225225,135135,45045,6435,24310,194480,680680,1361360,1701700,1361360,680680,194480,24310
%N A360651 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 A360651 G.f.: 2/(1 - 4*x + sqrt(1 - 4*x - 4*x*y) - 4*x*y).
%F A360651 T(n, k) = binomial(n, k)*CatalanNumber(n)*(2*n + 1). - _Peter Luschny_, Feb 15 2023
%e A360651 Triangle T(n, m) starts:
%e A360651 [0] 1;
%e A360651 [1] 3,     3;
%e A360651 [2] 10,    20,    10;
%e A360651 [3] 35,    105,   105,     35;
%e A360651 [4] 126,   504,   756,     504,     126;
%e A360651 [5] 462,   2310,  4620,    4620,    2310,    462;
%e A360651 [6] 1716,  10296, 25740,   34320,   25740,   10296,    1716;
%e A360651 [7] 6435,  45045, 135135,  225225,  225225,  135135,   45045,  6435;
%p A360651 CatalanNumber := n -> binomial(2*n, n)/(n + 1):
%p A360651 T := (n, k) -> (2*n + 1)*CatalanNumber(n)*binomial(n, k):
%p A360651 seq(seq(T(n, k), k = 0..n), n = 0..8); # _Peter Luschny_, Feb 15 2023
%o A360651 (Maxima)
%o A360651 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 A360651 Cf. A001700, A085880, A069720 (row sums).
%K A360651 nonn,tabl
%O A360651 0,2
%A A360651 _Vladimir Kruchinin_, Feb 15 2023