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Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n + 1, floor(k/2)).
3

%I #11 Mar 06 2023 09:51:59

%S 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,

%T 420,700,1,7,56,168,588,980,1960,1960,1,8,72,252,1008,2016,4704,5880,

%U 8820,1,9,90,360,1620,3780,10080,15120,26460,26460

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

%e Table T(n, k) starts:

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 2, 6;

%e [3] 1, 3, 12, 12;

%e [4] 1, 4, 20, 30, 60;

%e [5] 1, 5, 30, 60, 150, 150;

%e [6] 1, 6, 42, 105, 315, 420, 700;

%e [7] 1, 7, 56, 168, 588, 980, 1960, 1960;

%e [8] 1, 8, 72, 252, 1008, 2016, 4704, 5880, 8820;

%e [9] 1, 9, 90, 360, 1620, 3780, 10080, 15120, 26460, 26460.

%p A360857 := (n, k) -> binomial(n, ceil(k/2))*binomial(n + 1, floor(k/2)):

%p seq(seq(A360857(n, k), k=0..n), n=0..9);

%t 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 *)

%o (Python)

%o from math import comb

%o 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

%Y Cf. A360858, A360859.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Feb 28 2023