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

Author

R. H. Hardin, Aug 07 2012

Keywords

• nonn

Comments

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

Examples

			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
		

Links

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

Crossrefs

Column 3 of A215292.

Cf. A060854, A060693.

Programs

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

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 *)

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