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).

1, 3, 3, 10, 20, 10, 35, 105, 105, 35, 126, 504, 756, 504, 126, 462, 2310, 4620, 4620, 2310, 462, 1716, 10296, 25740, 34320, 25740, 10296, 1716, 6435, 45045, 135135, 225225, 225225, 135135, 45045, 6435, 24310, 194480, 680680, 1361360, 1701700, 1361360, 680680, 194480, 24310

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Formula

G.f.: 2/(1 - 4*x + sqrt(1 - 4*x - 4*x*y) - 4*x*y).

T(n, k) = binomial(n, k)*CatalanNumber(n)*(2*n + 1). - Peter Luschny, Feb 15 2023

Example

Triangle T(n, m) starts:
[0] 1;
[1] 3,     3;
[2] 10,    20,    10;
[3] 35,    105,   105,     35;
[4] 126,   504,   756,     504,     126;
[5] 462,   2310,  4620,    4620,    2310,    462;
[6] 1716,  10296, 25740,   34320,   25740,   10296,    1716;
[7] 6435,  45045, 135135,  225225,  225225,  135135,   45045,  6435;

Maple

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
T := (n, k) -> (2*n + 1)*CatalanNumber(n)*binomial(n, k):
seq(seq(T(n, k), k = 0..n), n = 0..8); # Peter Luschny, Feb 15 2023

Prog

(Maxima)
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);

Crossrefs

Cf. A001700, A085880, A069720 (row sums).

Sequence in context: A077899 A049973 A025519 * A076987 A229912 A321354

Adjacent sequences: A360648 A360649 A360650 * A360652 A360653 A360654

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Feb 15 2023

Status

approved