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%I #48 Sep 08 2022 08:44:52
%S 1,1,2,2,8,8,48,48,384,384,3840,3840,46080,46080,645120,645120,
%T 10321920,10321920,185794560,185794560,3715891200,3715891200,
%U 81749606400,81749606400,1961990553600,1961990553600,51011754393600,51011754393600,1428329123020800,1428329123020800
%N Number of solutions to non-attacking rooks problem on n X n board that are invariant under 180-degree rotation.
%C 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
%C 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
%C 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]
%D E. Lucas, Theorie des nombres, Gauthiers-Villars, Paris, 1891, Vol 1, p. 221.
%H Alois P. Heinz, <a href="/A037223/b037223.txt">Table of n, a(n) for n = 0..500</a>
%H E. Lucas, <a href="https://archive.org/details/thoriedesnombre00lucagoog/page/n245">Théorie des nombres</a>, Gauthiers-Villars, Paris, 1891, Vol 1, p. 221.
%H R. W. Robinson, <a href="http://dx.doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%H M. Szabo, <a href="http://www.nexus.hu/mikk/queen/index.html">Non-attacking Queens Problem Page</a>
%F a(2n) = a(2n+1) = n!*2^n.
%F 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
%F For asymptotics see the Robinson paper.
%F 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
%F 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
%F a(n) = Product_{i=1..floor(n/2)} 2*i. - _Wesley Ivan Hurt_, Oct 19 2014
%F 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
%p For Maple program see A000903.
%p # second Maple program:
%p a:= n-> (r-> r!*2^r)(iquo(n, 2)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Dec 23 2013
%t 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 *)
%t 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 *)
%o (Magma) [Factorial((n div 2) -1)*2^((n div 2)-1): n in [2..35]]; // _Vincenzo Librandi_, Nov 17 2018
%Y Cf. A000165, A033148, A037224, A032522, A037223.
%K nonn,easy
%O 0,3
%A Miklos SZABO (mike(AT)ludens.elte.hu)
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
%E Edited by _N. J. A. Sloane_, Sep 23 2006
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