A360282 Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.

1, 0, 1, 0, 3, 2, 0, 10, 12, 3, 0, 35, 60, 30, 4, 0, 126, 280, 210, 60, 5, 0, 462, 1260, 1260, 560, 105, 6, 0, 1716, 5544, 6930, 4200, 1260, 168, 7, 0, 6435, 24024, 36036, 27720, 11550, 2520, 252, 8

Offset: 0

Author

Peter Luschny, Feb 11 2023

Keywords

• nonn
• tabl

Comments

Inspired by Vladimir Kruchinin's A360546.

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     1;
[2] 0,     3,      2;
[3] 0,    10,     12,      3;
[4] 0,    35,     60,     30,      4;
[5] 0,   126,    280,    210,     60,     5;
[6] 0,   462,   1260,   1260,    560,   105,     6;
[7] 0,  1716,   5544,   6930,   4200,  1260,   168,    7;
[8] 0,  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;
[9] 0, 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;
		

Crossrefs

Cf. A135503, A088218 (and A001700), A182626 (row sums apart from sign).

Family: A172431 and unsigned A053123 (case 1), this sequence is case 2, A360546 (case 3).

Programs

Maple

T := proc(n, k) if n = 0 then 1 else m := n - k + 1; (1/2) * binomial(2*m, m) * binomial(m + n - 1, k - 1) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..8);
# With Vladimir Kruchinin's g.f.:
gf := 1 + (y - x*y^2)/(2*sqrt((x*y - 1)^2 - 4*x)) ;
serx := series(gf, x, 10): poly := n -> simplify(coeff(serx, x, n)):
seq(print(seq(coeff(poly(n), y, k), k = 0..n)), n = 0..9); # Peter Luschny, Feb 14 2023

Formula

The sequence is member of a family of sequences defined by T(0, 0, m) = 1 and for m > 0, n > 0 as T(n, k, m) = (1/m)*binomial(m*u, u)*binomial(u + n - 1, k - 1), where u = n - k + 1. Here the case m = 2 is considered.

G.f.: (y*(1 - x*y)) / (2*sqrt(x*(y*(x*y - 2) - 4) + 1)) - y/2 + 1. - Vladimir Kruchinin, Feb 14 2023