A226288 T(n,k) = [n/2]!*[(n+1)/2]!*C([n/2],k-1)*C([(n+1)/2],k-1) where [x] = floor(x).

1, 0, 1, 0, 1, 2, 0, 0, 4, 4, 0, 0, 0, 16, 12, 0, 0, 0, 4, 72, 36, 0, 0, 0, 0, 36, 324, 144, 0, 0, 0, 0, 0, 324, 1728, 576, 0, 0, 0, 0, 0, 36, 2592, 9216, 2880, 0, 0, 0, 0, 0, 0, 576, 20736, 57600, 14400, 0, 0, 0, 0, 0, 0, 0, 9216, 172800, 360000, 86400, 0, 0, 0, 0, 0, 0, 0, 576, 115200, 1440000, 2592000, 518400

Offset

1,6

Comments

T(n,k) = Number of permutations of n elements with 2k-2 odd displacements.

Links

R. H. Hardin, Table of n, a(n) for n = 1..10000

Sela Fried, Black-White Cell Capacity in k-ary Words and Permutations, arXiv:2509.07533 [math.CO], 2025. See p. 11.

Example

Table starts:
       1,        0,         0,         0,         0,        0
       1,        1,         0,         0,         0,        0
       2,        4,         0,         0,         0,        0
       4,       16,         4,         0,         0,        0
      12,       72,        36,         0,         0,        0
      36,      324,       324,        36,         0,        0
     144,     1728,      2592,       576,         0,        0
     576,     9216,     20736,      9216,       576,        0
    2880,    57600,    172800,    115200,     14400,        0
   14400,   360000,   1440000,   1440000,    360000,    14400
   86400,  2592000,  12960000,  17280000,   6480000,   518400
  518400, 18662400, 116640000, 207360000, 116640000, 18662400

Mathematica

T[n_, k_]:=(Floor[n/2])!*(Floor[(n+1)/2])!*Binomial[Floor[n/2], k-1]*Binomial[Floor[(n+1)/2], k-1]; Table[Reverse[Table[T[n-k+1, k], {k, n}]], {n, 12}]//Flatten (* Stefano Spezia, Jul 12 2024 *)

Crossrefs

Column 1 is A010551.

Columns 2-7 are: A226282-A226287.

Cf. A145891 (another version as irregular triangle).

Sequence in context: A108885 A341654 A072740 * A185146 A080964 A367054

Adjacent sequences: A226285 A226286 A226287 * A226289 A226290 A226291

Keyword

nonn,tabl

Author

R. H. Hardin, Jun 02 2013

Extensions

Connection of formula with combinatoric problem via N. J. A. Sloane in the Sequence Fans Mailing List, Jun 02 2013

Status

approved