A367267 Triangle read by rows. T(n, k) = binomial(n, k) * binomial(n - 1, k - 1).

1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 18, 12, 1, 0, 5, 40, 60, 20, 1, 0, 6, 75, 200, 150, 30, 1, 0, 7, 126, 525, 700, 315, 42, 1, 0, 8, 196, 1176, 2450, 1960, 588, 56, 1, 0, 9, 288, 2352, 7056, 8820, 4704, 1008, 72, 1, 0, 10, 405, 4320, 17640, 31752, 26460, 10080, 1620, 90, 1

Offset

0,5

Links

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

Formula

For k >= 2: T(n, k) = (n / k) * binomial(n-1, k-1)^2.

Example

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

Maple

T := (n, k) -> binomial(n, k) * binomial(n - 1, k - 1):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
# Or:
T := (n, k) -> if k=0 then k^n elif k=1 then n else (n/k)*binomial(n-1, k-1)^2 fi:
seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

A367267[n_, k_]:=Binomial[n, k]Binomial[n-1, k-1];
Table[A367267[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Nov 29 2023 *)

Crossrefs

Cf. A088218 (row sums), A367270 (row reversed).

Sequence in context: A350548 A105546 A342237 * A339030 A059297 A267222

Adjacent sequences: A367264 A367265 A367266 * A367268 A367269 A367270

Keyword

nonn,tabl

Author

Peter Luschny, Nov 11 2023

Status

approved