A215287 - OEIS

%I #14 Sep 08 2022 08:46:03

%S 1,3,10,30,140,420,2310,6930,42042,126126,816816,2450448,16628040,

%T 49884120,350574510,1051723530,7595781050,22787343150,168212023980,

%U 504636071940,3792416540640,11377249621920,86787993910800,260363981732400,2011383287449200

%N 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.

%C Also Schröder paths of length n having floor(n/2) peaks. - _Peter Luschny_, Sep 30 2018

%H R. H. Hardin, <a href="/A215287/b215287.txt">Table of n, a(n) for n = 1..210</a>

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

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

%F a(n) = (n - f + 1)*(2*n - f)! / ((n - f + 1)!^2 * f!) where f = floor(n/2). - _Peter Luschny_, Sep 30 2018

%e Some solutions for n=5:

%e   0 x 4   0 x 5   1 x 3   0 x 1   0 x 3   1 x 4   0 x 2

%e   x 3 x   x 1 x   x 0 x   x 4 x   x 2 x   x 0 x   x 1 x

%e   1 x 5   2 x 6   2 x 5   2 x 3   1 x 6   2 x 5   3 x 5

%e   x 7 x   x 3 x   x 6 x   x 6 x   x 5 x   x 6 x   x 6 x

%e   2 x 6   4 x 7   4 x 7   5 x 7   4 x 7   3 x 7   4 x 7

%p T := (n, k) -> (n-k+1)*(2*n-k)!/((n-k+1)!^2*k!):

%p a := n -> T(n, floor(n/2)): seq(a(n), n = 1..23); # _Peter Luschny_, Sep 30 2018

%t 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 *)

%o (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

%Y Column 3 of A215292.

%Y Cf. A060854, A060693.

%K nonn

%O 1,2

%A _R. H. Hardin_, Aug 07 2012