[seqfan] Re: A series for exp(Pi)

[Maximilian Hasler]

On Sun, Oct 25, 2009 at 6:10 AM, Jaume Oliver i Lafont
<joliverlafont at gmail.com> wrote:
> these one-line steps compute the sequence on PARI:
> gp > G(n)=gamma(n)
> gp > even(n)=(1+(-1)^n)/G(3*I)/G(-3*I)
> gp > odd(n)=3*(1-(-1)^n)/G(1/2+3*I)/G(1/2-3*I)
> gp > a(n)=2^(n-1)*G(n/2+3*I)*G(n/2-3*I)*(even(n)+odd(n))
>
> Six gamma functions with different argument appear. Is there a simpler solution?

You may observe that Gamma(z) always occurs multiplied with
Gamma(conjugate(z)), thus the above straightforwardly simplifies to:

H(x)=norm(gamma(x))
a(n)=round(2^n*H(n/2+3*I)/if(n%2,H(1/2+3*I)/3,H(3*I)))

and with Richard's remark:
a(n)=round(2^n*norm(gamma(n/2+3*I))*if(n%2,sinh(3*Pi),cosh(3*Pi))/Pi*3)

vector(20,n,a(n-1))
= [1, 6, 36, 222, 1440, 9990, 74880, 609390, 5391360, 51798149,
539136007, 6060383471, 73322496955, 951480204957, 13198049451902,
195053442016213, 3061947472841328, 50908948366231639,
894088662069667631, 16545408219025282783]

Maximilian