A293857 a(n) is the number of permutations {c_1..c_n} of {1..n} for which c_1 - c_2 + ... + (-1)^(n-1)*c_n are squares.

1, 1, 1, 4, 12, 36, 144, 1440, 9216, 66240, 504000, 7344000, 73612800, 830995200, 9373190400, 181875456000, 2474319052800, 38246274662400, 572552876851200, 13783143886848000, 237527801118720000, 4658378696294400000, 86818505051013120000, 2488457229932298240000

Offset

0,4

Comments

For a permutation C = {c_1..c_n} of {1..n}, set s(C) = c_1 - c_2 + ... + (-1)^(n-1)*c_n. Then max s(C) is square that is (ceil(n/2))^2 or A008794(n+1).

a(n)/n! is slowly and non-monotonically decreasing: 1, 1/2, 2/3, 1/2, 3/10, 1/5, 2/7, 8/35, 23/126, 5/36, 85/462, 71/462, ... .

Positions for which a(n) divisible by all primes <= n: 1, 4, 10, ... .

The smallest primes <= n not dividing a(n) or 0 if there is no such primes: 0, 2, 3, 0, 5, 5, 7, 5, 7, 0, 7, 7, ... .

Let k = floor(n / 2). Then a(n) = divisible by k! * (n-k)!. - David A. Corneth, Oct 18 2017. (For a proof, cf. comment in A293984. - Vladimir Shevelev, Nov 06 2017)

Links

Peter J. C. Moses, Table of n, a(n) for n = 0..200

Formula

From author's comment in A008794 it follows that a(n) >= A010551(n).

Example

Let n=3. For a permutation C={c_1,c_2,c_3}, set s = s(C) = c_1 - c_2 + c_3. We have the permutations:
1,2,3; s=2
1,3,2; s=0
2,1,3; s=4
2,3,1; s=0
3,1,2; s=4
3,2,1; s=2
Here there are 4 permutations for which {s} are squares. So a(3)=4.

Maple

b:= proc(p, m, s) option remember; (n-> `if`(n=0, `if`(issqr(s), 1, 0),
      `if`(p>0, b(p-1, m, s+n), 0)+`if`(m>0, b(p, m-1, s-n), 0)))(p+m)
    end:
a:= n-> (t-> b(n-t, t, 0)*t!*(n-t)!)(iquo(n, 2)):
seq(a(n), n=0..28);  # Alois P. Heinz, Sep 17 2020

Mathematica

a293857=Table[Total[(Floor[n/2]!*(n-Floor[n/2])!)(Reverse[Map[SeriesCoefficient[QBinomial[n, Floor[(n+1)/2], q], {q, 0, #}]&, Map[2#(Floor[(n+1)/2] - #)&, Range[0, Floor[(n+1)/4]]]]]
)], {n, 25}] (* Peter J. C. Moses, Nov 01 2017 *)

Crossrefs

Cf. A000290, A008794, A293783, A293984.

Sequence in context: A192010 A252697 A056383 * A052643 A140123 A164853

Adjacent sequences: A293854 A293855 A293856 * A293858 A293859 A293860

Keyword

nonn

Author

Vladimir Shevelev, Oct 17 2017

Extensions

a(5)-a(12) from Peter J. C. Moses, Oct 17 2017

a(13)-a(23) from David A. Corneth, Oct 17 2017

a(0)=1 prepended by Alois P. Heinz, Sep 17 2020

Status

approved