A370232 Triangle read by rows. T(n, k) = binomial(n + k, 2*k)^2.

1, 1, 1, 1, 9, 1, 1, 36, 25, 1, 1, 100, 225, 49, 1, 1, 225, 1225, 784, 81, 1, 1, 441, 4900, 7056, 2025, 121, 1, 1, 784, 15876, 44100, 27225, 4356, 169, 1, 1, 1296, 44100, 213444, 245025, 81796, 8281, 225, 1, 1, 2025, 108900, 853776, 1656369, 1002001, 207025, 14400, 289, 1

Offset: 0

Author

Peter Luschny, Feb 12 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 1,     1;
[2] 1,     9,       1;
[3] 1,    36,      25,       1;
[4] 1,   100,     225,      49,       1;
[5] 1,   225,    1225,     784,      81,      1;
[6] 1,   441,    4900,    7056,    2025,    121,     1;
[7] 1,   784,   15876,   44100,   27225,   4356,   169,   1;
		

Crossrefs

Shifted bisection of A182878.

Cf. A370233 (c=2), A188648 (row sums), A188662 (central terms).

Programs

Mathematica

Table[Binomial[n + k, 2*k]^2, {n, 0, 7}, {k, 0, n}] // Flatten

Formula

T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n + k, 2*k) * Pochhammer(n - k + c, 2*k) * z^k / (2*k)! and c = 1.

T(n, k) = [z^k] hypergeom([-n, -n, 1 + n, 1 + n], [1/2, 1/2, 1], z/16).