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%I #29 Sep 08 2022 08:45:29
%S -1,1,9,75,735,8505,114345,1756755,30405375,585810225,12439852425,
%T 288735522075,7273385294175,197646339515625,5763367260275625,
%U 179518217255251875,5948862302837829375,208977775735174070625,7757508341684492015625,303429397707601987696875
%N a(n) = (2*n)!*(2*n-1)/(2^n*n!).
%D V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.
%H G. C. Greubel, <a href="/A126965/b126965.txt">Table of n, a(n) for n = 0..250</a>
%F E.g.f.: sqrt(1-4*x)/(1-2*x).
%F G.f.: x - 1 + 9*x^2/(Q(0)-9*x), where Q(k)= 1 + 9*x + 2*k*(1+6*x) + 4*x*k^2 - x*(2*k+1)*(2*k+5)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Apr 25 2013
%F a(n) = (1/sqrt(Pi)) * Numerator(Gamma((2n+3)/2) - Gamma((2n+1)/2)), for n>=0. Denominators are 2^(n+1). - _Richard R. Forberg_, Feb 22 2015
%F +(-2*n+3)*a(n) +(2*n-1)^2*a(n-1)=0. - _R. J. Mathar_, Jun 17 2016
%p seq( ((2*n)!*(2*n-1))/(2^n*n!), n=0..20); # _G. C. Greubel_, Jan 29 2020
%t Table[((2n)!(2n-1))/(2^n n!),{n,0,20}] (* _Harvey P. Dale_, Jan 16 2017 *)
%o (PARI) vector(21, n, my(m=n-1); ((2*m)!*(2*m-1))/(2^m*m!)) \\ _G. C. Greubel_, Mar 19 2017
%o (PARI) apply( {A126965(n)=(2*n)!*(2*n-1)/(2^n*n!)}, [0..20]) \\ _M. F. Hasler_, Feb 27 2020
%o (Magma) F:=Factorial; [(F(2*n)*(2*n-1))/(2^n*F(n)): n in [0..20]]; // _G. C. Greubel_, Jan 29 2020
%o (Sage) f=factorial; [(f(2*n)*(2*n-1))/(2^n*f(n)) for n in (0..20)] # _G. C. Greubel_, Jan 29 2020
%o (GAP) F:=Factorial;; List([0..20], n-> (F(2*n)*(2*n-1))/(2^n*F(n)) ); # _G. C. Greubel_, Jan 29 2020
%Y Cf. A001147.
%K sign
%O 0,3
%A _N. J. A. Sloane_, Mar 21 2007