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A135457 a(n) = (2n-1)!! * Sum_{k=0..n-2}(-1)^k/(2k+1).

%I A135457 #45 Sep 08 2022 08:45:32
%S A135457 0,3,10,91,684,8679,100542,1664055,25991640,532354635,10455799410,
%T A135457 255542155155,6044821114500,171748491958575,4751436512960550,
%U A135457 153911731348760175,4874807783839316400,177334729873063945875
%N A135457 a(n) = (2n-1)!! * Sum_{k=0..n-2}(-1)^k/(2k+1).
%H A135457 Vincenzo Librandi, <a href="/A135457/b135457.txt">Table of n, a(n) for n = 1..200</a>
%H A135457 Peter Luschny, <a href="http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html">Is the Gamma function misdefined? Hadamard versus Euler - Who found the better Gamma function?</a>
%F A135457 a(n) = (-1/4)(Product_{i=1..n}(2i-3))((2n-1)Pi + 2(-1)^n*Sum_{k>=0}k!/ Product_{j=1..k}(2j+2n-1)). - _Benoit Cloitre_, Dec 15 2007
%F A135457 a(n+3) = 4*a(n+2) + (4n^2+12n+1)*a(n+1) - (8n^2-2)*a(n) with a(1)=0, a(2)=3, a(3)=10. - _Benoit Cloitre_, Dec 15 2007
%F A135457 a(n) ~ Pi * 2^(n-3/2) * n^n / exp(n). - _Vaclav Kotesovec_, Oct 11 2013
%F A135457 a(n+1) = (2n+1)*(a(n) - (-1)^n (2n-3)!!) with a(1)=0. - _Cyril Damamme_, Jul 16 2015
%F A135457 a(n) = (2^(n-2)*Gamma(n+1/2)*((-1)^n*(Psi(n/2+1/4)-Psi(n/2-1/4))+Pi))/sqrt(Pi). - _Peter Luschny_, Jul 18 2015
%F A135457 a(n) = A167576(n) - A024199(n). - _Cyril Damamme_, Jul 22 2015
%p A135457 a := n -> (2^(n-2)*GAMMA(n+1/2)*((-1)^n*(Psi(n/2+1/4)-Psi(n/2-1/4))+Pi))/sqrt(Pi);
%p A135457 seq(a(n), n=1..18); # _Peter Luschny_, Jul 18 2015
%t A135457 FullSimplify[Table[(2^(n-2)*(n-1/2)!*(Pi+2*(-1)^n*LerchPhi[-1,1,n-1/2]))/Sqrt[Pi],{n,1,20}]] (* _Vaclav Kotesovec_, Oct 11 2013 *)
%o A135457 (PARI) a(n)=round((-1/4)*prod(i=1,n,2*i-3)*(Pi*(2*n-1)+2*(-1)^n*sum(k=0,1500,1.*k!/prod(i=1,k,(2*i+2*n-1)))))
%o A135457 (Magma) I:=[0,3,10]; [n le 3 select I[n] else 4*Self(n-1)+(4*n^2-12*n+1)*Self(n-2)-(8*n^2-48*n+70)*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jul 17 2015
%Y A135457 Cf. A167576 and A024199.
%K A135457 nonn
%O A135457 1,2
%A A135457 _Benoit Cloitre_, Dec 15 2007
%E A135457 Definition replaced by a simplified one by _Cyril Damamme_, Jul 18 2015