a := n -> denom(GAMMA(n+1/2)^2/(2*n*Pi)):1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
b := n -> 2^(2*n+1+padic:-ordp(n,2)):
seq(round(evalf(a(n)/b(n),800)), n=1..290);
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
For n::posint larger than 100 the call GAMMA(n) returns unevaluated.
[...] With that in mind your use of `denom` above in procedure `a`
presumes a particular form that you might have realized would not attain.
On Saturday, September 23, 2017 at 9:46:35 AM UTC-4, Peter Luschny wrote:1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
a := n -> denom(GAMMA(n+1/2)^2/(2*n*Pi)):
b := n -> 2^(2*n+1+padic:-ordp(n,2)):
seq(round(evalf(a(n)/b(n),800)), n=1..290);
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
For n::posint larger than 100 the call GAMMA(n) returns unevaluated. (You'd likely have discover this if you didn't wrap with `denom`, to look at the intermediate result, BTW.) The help page ?GAMMA indicates that the `expand` command can deal with thatcase. But for n=201/2 a rational that doesn't do the job.
From Maple 18 onwards, this formula could be obtained:
lprint( convert(GAMMA(n), doublefactorial) );
2*doublefactorial(2*n-2)*(2/Pi)^(-1/4+1/4*cos(2*Pi*n))/(2^n)
And that `doublefactorial` could also be "expanded", say with further conversion to `factorial`.
ee:=GAMMA(101+1/2);
ee := GAMMA(203/2)
ff:=convert(convert(ee,doublefactorial),factorial):
lprint(%);
13399280426846637026266542163537418668282223673245521\ 36009000175903930373973956200201339808971908813302485\ 337859343555595880232757068702494068539027184400769291\ 974725364464819431304931640625/25353012004564588029934\
06410752/(1/Pi)^(1/2)
It's interesting that this call to `simplify` works, even if conversion of `ee` is a more obscure task.
simplify(ee - ff);
0
If your Maple version is even older, and does not provide that conversion formula of GAMMA(n) to that formula involving doublefactorial, here it is:
lprint( convert(GAMMA(n), doublefactorial) ); 2*doublefactorial(2*n-2)*(2/Pi)^(-1/4+1/4*cos(2*Pi*n))/(2^n)
In modern Maple then, one could do this:
a := n -> GAMMA(n+1/2)^2/(2*n*Pi):
b := n -> 2^(2*n+1+padic:-ordp(n,2)):
c := n -> convert(convert(GAMMA(n+1/2),doublefactorial),factorial)^2/(2*n*Pi):
seq( simplify(a(n)-c(n)), n=98..101 );
0, 0, 0, 0
seq(denom(c(n))/b(n), n=98..101);
1, 1, 1, 1
seq(denom(c(n))/b(n), n=1..290, 10);
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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