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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
1, 3, 6, 30, 70, 420, 1050, 6930, 18018, 126126, 336336, 2450448, 6651216, 49884120, 137181330, 1051723530, 2921454250, 22787343150, 63804560820, 504636071940, 1422156202740, 11377249621920, 32235540595440, 260363981732400 (list; graph; refs; listen; history; text; internal format)
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: A136946 A125521 A211168 * A090932 A280981 A265376

Adjacent sequences:  A215291 A215292 A215293 * A215295 A215296 A215297

KEYWORD

nonn

AUTHOR

R. H. Hardin, Aug 07 2012

STATUS

approved

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Last modified November 15 03:42 EST 2018. Contains 317225 sequences. (Running on oeis4.)