A360858 Triangle read by rows. T(n, k) = binomial(n + 1, ceil(k/2)) * binomial(n, floor(k/2)).

1, 1, 2, 1, 3, 6, 1, 4, 12, 18, 1, 5, 20, 40, 60, 1, 6, 30, 75, 150, 200, 1, 7, 42, 126, 315, 525, 700, 1, 8, 56, 196, 588, 1176, 1960, 2450, 1, 9, 72, 288, 1008, 2352, 4704, 7056, 8820, 1, 10, 90, 405, 1620, 4320, 10080, 17640, 26460, 31752

Offset

0,3

Links

Table of n, a(n) for n=0..54.

Example

Triangle T(n, k) starts:
[0] 1;
[1] 1,  2;
[2] 1,  3,  6;
[3] 1,  4, 12,  18;
[4] 1,  5, 20,  40,   60;
[5] 1,  6, 30,  75,  150,  200;
[6] 1,  7, 42, 126,  315,  525,   700;
[7] 1,  8, 56, 196,  588, 1176,  1960,  2450;
[8] 1,  9, 72, 288, 1008, 2352,  4704,  7056,  8820;
[9] 1, 10, 90, 405, 1620, 4320, 10080, 17640, 26460, 31752.

Maple

A360858 := (n, k) -> binomial(n + 1, ceil(k/2))*binomial(n, floor(k/2)):
seq(seq(A360858(n, k), k = 0..n), n = 0..9);

Prog

(Python)
from math import comb
def A360858_T(n, k): return comb(n, m:=k>>1)**2*(n+1)//(m+1 if k&1 else n+1-m) # Chai Wah Wu, Feb 28 2023

Crossrefs

Cf. A005566 (main diagonal), A001700 (row sums).

Cf. A360857, A360859.

Sequence in context: A210238 A209936 A213941 * A181511 A115196 A093346

Adjacent sequences: A360855 A360856 A360857 * A360859 A360860 A360861

Keyword

nonn,tabl

Author

Peter Luschny, Feb 28 2023

Status

approved