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, 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

Links

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

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 * A361893 A244129 A363907

Adjacent sequences: A350263 A350264 A350265 * A350267 A350268 A350269

Keyword

nonn,tabl

Author

Peter Luschny, Jan 09 2022

Status

approved