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A037223
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Number of solutions to non-attacking rooks problem on n X n board that are invariant under 180-degree rotation.
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15
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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E. Lucas, Theorie des nombres, Gauthiers-Villars, Paris, 1891, Vol 1, p. 221.
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LINKS
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FORMULA
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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
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
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MAPLE
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# second Maple program:
a:= n-> (r-> r!*2^r)(iquo(n, 2)):
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MATHEMATICA
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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 *)
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PROG
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(Magma) [Factorial((n div 2) -1)*2^((n div 2)-1): n in [2..35]]; // Vincenzo Librandi, Nov 17 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Miklos SZABO (mike(AT)ludens.elte.hu)
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
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STATUS
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approved
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