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

0, 3, 10, 91, 684, 8679, 100542, 1664055, 25991640, 532354635, 10455799410, 255542155155, 6044821114500, 171748491958575, 4751436512960550, 153911731348760175, 4874807783839316400, 177334729873063945875

Offset: 1

Author

Benoit Cloitre, Dec 15 2007

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Peter Luschny, Is the Gamma function misdefined? Hadamard versus Euler - Who found the better Gamma function?

Crossrefs

Cf. A167576 and A024199.

Programs

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

Maple

a := n -> (2^(n-2)*GAMMA(n+1/2)*((-1)^n*(Psi(n/2+1/4)-Psi(n/2-1/4))+Pi))/sqrt(Pi);
seq(a(n), n=1..18); # Peter Luschny, Jul 18 2015

Mathematica

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 *)

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)))))

Formula

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

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

a(n) ~ Pi * 2^(n-3/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 11 2013

a(n+1) = (2n+1)*(a(n) - (-1)^n (2n-3)!!) with a(1)=0. - Cyril Damamme, Jul 16 2015

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

a(n) = A167576(n) - A024199(n). - Cyril Damamme, Jul 22 2015

Extensions

Definition replaced by a simplified one by Cyril Damamme, Jul 18 2015