A215294 Number of permutations of 0..floor((n*3-2)/2) on odd squares of an n X 3 array such that each row and column of odd squares is increasing.

1, 3, 6, 30, 70, 420, 1050, 6930, 18018, 126126, 336336, 2450448, 6651216, 49884120, 137181330, 1051723530, 2921454250, 22787343150, 63804560820, 504636071940, 1422156202740, 11377249621920, 32235540595440, 260363981732400

Offset: 1

Author

R. H. Hardin, Aug 07 2012

Keywords

• nonn

Comments

a(n) is number of symmetric standard Young tableaux of shape (n,n,n). - Ran Pan, May 21 2015

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• Ran Pan, Problem 4, Project P.

Crossrefs

Column 3 of A215297.

Cf. A060693.

Programs

Maple

a := n -> `if`(irem(n, 2) = 0, ((1/2)*n+1)*factorial((3/2)*n)/ (factorial((1/2)*n+1)^2*factorial((1/2)*n)), factorial((3/2)*n+3/2)/ (factorial((1/2)*n+1/2)^3*((9/2)*n+3/2))): # Peter Luschny, Sep 30 2018

Formula

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

a(n) = e(n) if n even otherwise o(n), where e(n) = 6*Gamma((3*n)/2)/((2 + n)*Gamma(1 + n/2)^2*Gamma(n/2)) and o(n) = (1 + n)*Gamma(1/2 + (3*n)/2)/(2*Gamma((3 + n)/2)^3). - Peter Luschny, Sep 30 2018