A360859 Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n, floor(k/2)).

1, 1, 1, 1, 2, 4, 1, 3, 9, 9, 1, 4, 16, 24, 36, 1, 5, 25, 50, 100, 100, 1, 6, 36, 90, 225, 300, 400, 1, 7, 49, 147, 441, 735, 1225, 1225, 1, 8, 64, 224, 784, 1568, 3136, 3920, 4900, 1, 9, 81, 324, 1296, 3024, 7056, 10584, 15876, 15876, 1, 10, 100, 450, 2025, 5400, 14400, 25200, 44100, 52920, 63504

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Example

Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2,  4;
[3] 1, 3,  9,   9;
[4] 1, 4, 16,  24,   36;
[5] 1, 5, 25,  50,  100,  100;
[6] 1, 6, 36,  90,  225,  300,  400;
[7] 1, 7, 49, 147,  441,  735, 1225,  1225;
[8] 1, 8, 64, 224,  784, 1568, 3136,  3920,  4900;
[9] 1, 9, 81, 324, 1296, 3024, 7056, 10584, 15876, 15876;

Maple

A360859 := (n, k) -> binomial(n, ceil(k/2)) * binomial(n, floor(k/2)):
seq(seq(A360859(n, k), k = 0..n), n = 0..10);

Prog

(Python)
from math import comb
def A360859_T(n, k): return comb(n, m:=k>>1)**2*(n-m)//(m+1) if k&1 else comb(n, k>>1)**2 # Chai Wah Wu, Feb 28 2023

Crossrefs

Cf. A018224 (main diagonal), A360861 (row sums).

Cf. A360857, A360858.

Sequence in context: A275895 A369292 A158613 * A209573 A100075 A059836

Adjacent sequences: A360856 A360857 A360858 * A360860 A360861 A360862

Keyword

nonn,tabl

Author

Peter Luschny, Feb 28 2023

Status

approved