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

1,2

Comments

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

Links

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

Ran Pan, Problem 4, Project P.

Formula

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

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

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

Example

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

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

Crossrefs

Column 3 of A215297.

Cf. A060693.

Sequence in context: A125521 A211168 A355989 * A350756 A090932 A361864

Adjacent sequences: A215291 A215292 A215293 * A215295 A215296 A215297

Keyword

nonn

Author

R. H. Hardin, Aug 07 2012

Status

approved