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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Column 3 of A215292.

Cf. A060854, A060693.

Sequence in context: A136846 A004663 A136853 * A316764 A331780 A290061

Adjacent sequences:  A215284 A215285 A215286 * A215288 A215289 A215290

KEYWORD

nonn

AUTHOR

R. H. Hardin, Aug 07 2012

STATUS

approved

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Last modified October 23 10:54 EDT 2021. Contains 348211 sequences. (Running on oeis4.)