A037223 Number of solutions to non-attacking rooks problem on n X n board that are invariant under 180-degree rotation.

1, 1, 2, 2, 8, 8, 48, 48, 384, 384, 3840, 3840, 46080, 46080, 645120, 645120, 10321920, 10321920, 185794560, 185794560, 3715891200, 3715891200, 81749606400, 81749606400, 1961990553600, 1961990553600, 51011754393600, 51011754393600, 1428329123020800, 1428329123020800

Offset: 0

Author

Miklos SZABO (mike(AT)ludens.elte.hu)

Keywords

• nonn
• easy

Comments

This is just A000165 doubled up. Normally such sequences do not get their own entry in the OEIS. This is an exception. - N. J. A. Sloane, Sep 23 2006

Also the number of permutations of (1,2,3,...,n) for which the reverse of the inverse is the same as the inverse of the reverse. - Ian Duff, Mar 09 2007

Conjecture: a(n) = Product_{1<=i<=n and phi(i)<=floor(i/2)}i. - Enrique Pérez Herrero, May 31 2012. This conjecture is WRONG, counterexample is n=105. [Vaclav Kotesovec, Sep 07 2012]

References

• E. Lucas, Theorie des nombres, Gauthiers-Villars, Paris, 1891, Vol 1, p. 221.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• E. Lucas, Théorie des nombres, Gauthiers-Villars, Paris, 1891, Vol 1, p. 221.
• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
• M. Szabo, Non-attacking Queens Problem Page

Crossrefs

Cf. A000165, A033148, A037224, A032522, A037223.

Programs

Magma

[Factorial((n div 2) -1)*2^((n div 2)-1): n   in [2..35]]; // Vincenzo Librandi, Nov 17 2018

Maple

For Maple program see A000903.
# second Maple program:
a:= n-> (r-> r!*2^r)(iquo(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 23 2013

Mathematica

f[n_]:=Times@@Select[Range[n],EulerPhi[#]<=Floor[#/2]&]; Table[f[n],{n,1,30}] (* Conjectured: Enrique Pérez Herrero, May 31 2012 *)(* This conjecture and also program is WRONG for n=105, Vaclav Kotesovec, Sep 07 2012 *)
a[n_] := (2*Floor[n/2])!!; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Dec 23 2013, after N. J. A. Sloane's comment *)

Formula

a(2n) = a(2n+1) = n!*2^n.

E.g.f.: 1 + x + (1 + x + x^2)*exp(x^2/2)*sqrt(Pi/2)*erf(x/sqrt(2)), where erf denotes the error function. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

For asymptotics see the Robinson paper.

E.g.f.: Q(0) where Q(k)= 1 + x/(2*k + 1 - x*(2*k+1)/(x+1/Q(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 21 2012

E.g.f.: 1/(W(0)-x) where W(k)= x + 1/(1 + x/(2*k + 1 - x*(2*k+1)/W(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 22 2012

a(n) = Product_{i=1..floor(n/2)} 2*i. - Wesley Ivan Hurt, Oct 19 2014

D-finite with recurrence: a(n) +a(n-1) -n*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Feb 20 2020

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

Edited by N. J. A. Sloane, Sep 23 2006