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A367270
Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
3
1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 18, 4, 0, 1, 20, 60, 40, 5, 0, 1, 30, 150, 200, 75, 6, 0, 1, 42, 315, 700, 525, 126, 7, 0, 1, 56, 588, 1960, 2450, 1176, 196, 8, 0, 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 0, 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10, 0
OFFSET
0,5
FORMULA
For 0< k < n: T(n, k) = ((n - k) / n)*binomial(n, k)^2.
EXAMPLE
Triangle T(n, k) begins:
[0] 1;
[1] 1, 0;
[2] 1, 2, 0;
[3] 1, 6, 3, 0;
[4] 1, 12, 18, 4, 0;
[5] 1, 20, 60, 40, 5, 0;
[6] 1, 30, 150, 200, 75, 6, 0;
[7] 1, 42, 315, 700, 525, 126, 7, 0;
[8] 1, 56, 588, 1960, 2450, 1176, 196, 8, 0;
[9] 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 0;
MAPLE
T := (n, k) -> binomial(n, k) * binomial(n - 1, n - k - 1):
# Or:
T := (n, k) -> if k=0 then 1 elif k=n then 0 else ((n-k)/n)*binomial(n, k)^2 fi:
seq(seq(T(n, k), k = 0..n), n = 0..9);
MATHEMATICA
A367270[n_, k_]:=Binomial[n, k]Binomial[n-1, n-k-1];
Table[A367270[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Nov 29 2023 *)
CROSSREFS
Cf. A088218 (row sums), A367267 (row reversed).
Sequence in context: A367795 A343825 A339031 * A365770 A059299 A332673
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 11 2023
STATUS
approved