A360857 Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n + 1, floor(k/2)).

1, 1, 1, 1, 2, 6, 1, 3, 12, 12, 1, 4, 20, 30, 60, 1, 5, 30, 60, 150, 150, 1, 6, 42, 105, 315, 420, 700, 1, 7, 56, 168, 588, 980, 1960, 1960, 1, 8, 72, 252, 1008, 2016, 4704, 5880, 8820, 1, 9, 90, 360, 1620, 3780, 10080, 15120, 26460, 26460

Offset: 0

Author

Peter Luschny, Feb 28 2023

Keywords

• nonn
• tabl

Examples

			Table T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2,  6;
[3] 1, 3, 12,  12;
[4] 1, 4, 20,  30,   60;
[5] 1, 5, 30,  60,  150,  150;
[6] 1, 6, 42, 105,  315,  420,   700;
[7] 1, 7, 56, 168,  588,  980,  1960,  1960;
[8] 1, 8, 72, 252, 1008, 2016,  4704,  5880,  8820;
[9] 1, 9, 90, 360, 1620, 3780, 10080, 15120, 26460, 26460.
		

Crossrefs

Cf. A360858, A360859.

Programs

Maple

A360857 := (n, k) -> binomial(n, ceil(k/2))*binomial(n + 1, floor(k/2)):
seq(seq(A360857(n, k), k=0..n), n=0..9);

Mathematica

Table[Binomial[n,Ceiling[k/2]]Binomial[n+1,Floor[k/2]],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 06 2023 *)

Python

from math import comb
def A360857_T(n,k): return comb(n+1,m:=k>>1)**2*(n+1-m)*(n-m)//((m+1)*(n+1)) if k&1 else comb(n+1,m:=k>>1)**2*(n+1-m)//(n+1) # Chai Wah Wu, Feb 28 2023