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A350266
Triangle read by rows. T(n, k) = binomial(n, k) * n! / (n - k + 1)! if k >= 1, if k = 0 then T(n, k) = k^n. T(n, k) for 0 <= k <= n.
1
1, 0, 1, 0, 2, 2, 0, 3, 9, 6, 0, 4, 24, 48, 24, 0, 5, 50, 200, 300, 120, 0, 6, 90, 600, 1800, 2160, 720, 0, 7, 147, 1470, 7350, 17640, 17640, 5040, 0, 8, 224, 3136, 23520, 94080, 188160, 161280, 40320, 0, 9, 324, 6048, 63504, 381024, 1270080, 2177280, 1632960, 362880
OFFSET
0,5
FORMULA
T(n, k) = binomial(n, k)^2 * k! / (n - k + 1) if k >= 1.
EXAMPLE
Table starts:
[0] 1;
[1] 0, 1;
[2] 0, 2, 2;
[3] 0, 3, 9, 6;
[4] 0, 4, 24, 48, 24;
[5] 0, 5, 50, 200, 300, 120;
[6] 0, 6, 90, 600, 1800, 2160, 720;
[7] 0, 7, 147, 1470, 7350, 17640, 17640, 5040;
[8] 0, 8, 224, 3136, 23520, 94080, 188160, 161280, 40320;
[9] 0, 9, 324, 6048, 63504, 381024, 1270080, 2177280, 1632960, 362880;
MAPLE
T := (n, k) -> ifelse(k = 0, k^n, binomial(n, k)^2 * k! / (n - k + 1)):
seq(seq(T(n, k), k = 0..n), n = 0..9);
MATHEMATICA
T[n_, 0] := Boole[n == 0]; T[n_, k_] := Binomial[n, k]^2 * k!/(n - k + 1); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jan 09 2022 *)
CROSSREFS
A350267 (row sums), A000142 (main diagonal), A074143 (subdiagonal), A006002 (column 2), A089835 (central terms).
Sequence in context: A336978 A011137 A143396 * A376724 A375470 A361893
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 09 2022
STATUS
approved