A215287 Number of permutations of 0..floor((n*3-1)/2) on even squares of an n X 3 array such that each row and column of even squares is increasing.

1, 3, 10, 30, 140, 420, 2310, 6930, 42042, 126126, 816816, 2450448, 16628040, 49884120, 350574510, 1051723530, 7595781050, 22787343150, 168212023980, 504636071940, 3792416540640, 11377249621920, 86787993910800, 260363981732400, 2011383287449200

Offset

1,2

Comments

Also Schröder paths of length n having floor(n/2) peaks. - Peter Luschny, Sep 30 2018

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Formula

f3 = floor((n+1)/2); f4 = floor(n/2);

a(n) = A060854(2,f3)*A060854(1,f4)*binomial(2*f3+1*f4,2*f3).

a(n) = (n - f + 1)*(2*n - f)! / ((n - f + 1)!^2 * f!) where f = floor(n/2). - Peter Luschny, Sep 30 2018

Example

Some solutions for n=5:
  0 x 4   0 x 5   1 x 3   0 x 1   0 x 3   1 x 4   0 x 2
  x 3 x   x 1 x   x 0 x   x 4 x   x 2 x   x 0 x   x 1 x
  1 x 5   2 x 6   2 x 5   2 x 3   1 x 6   2 x 5   3 x 5
  x 7 x   x 3 x   x 6 x   x 6 x   x 5 x   x 6 x   x 6 x
  2 x 6   4 x 7   4 x 7   5 x 7   4 x 7   3 x 7   4 x 7

Maple

T := (n, k) -> (n-k+1)*(2*n-k)!/((n-k+1)!^2*k!):
a := n -> T(n, floor(n/2)): seq(a(n), n = 1..23); # Peter Luschny, Sep 30 2018

Mathematica

Table[(n - Floor[n/2] + 1) (2 n - Floor[n/2])! / ((n -Floor[n/2] + 1)!^2 Floor[n/2]!), {n, 1, 30}] (* Vincenzo Librandi, Oct 01 2018 *)

Prog

(Magma) [(n-(n div 2)+1)*Factorial(2*n-(n div 2)) / (Factorial(n-(n div 2) +1)^2*Factorial((n div 2))): n in [1..30]]; // Vincenzo Librandi, Oct 01 2018

Crossrefs

Column 3 of A215292.

Cf. A060854, A060693.

Sequence in context: A004663 A136853 A361911 * A316764 A331780 A360563

Adjacent sequences: A215284 A215285 A215286 * A215288 A215289 A215290

Keyword

nonn

Author

R. H. Hardin, Aug 07 2012

Status

approved