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Numerator of binomial(2*n,n)/20^n.
2

%I #32 Mar 11 2024 01:50:29

%S 1,1,3,1,7,63,231,429,1287,2431,46189,88179,676039,52003,200583,

%T 1938969,60108039,116680311,90751353,176726319,6892326441,13456446861,

%U 52602474093,20583576819,322476036831,15801325804719,61989816618513,121683714103007,191217265019011,375840831244263,7391536347803839

%N Numerator of binomial(2*n,n)/20^n.

%C a(n) is for p=1, q=5. Generally for p,q in N, p>0, q>1:

%C Sum_{k>=0}(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=sqrt(q/(q-p)).

%C Sum_{k>=0}(-1)^k*(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=sqrt(q/(q+p)).

%C Sum_{k>=0}(-1)^(k+1)*(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=-sqrt(q/(q+p)).

%C a(n) is the numerator of binomial(2*n,n)/20^n. - _Robert Israel_, Apr 09 2017

%H Chai Wah Wu, <a href="/A285020/b285020.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n)/A285021(n) = (-1/5)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).

%F Sum_{k>=0} a(k)/A285021(k) = sqrt(5)/2.

%F Sum_{k>=0} (-1)^k*a(k)/A285021(k) = sqrt(5/6).

%F Sum_{k>=0} (-1)^(k+1)*a(k)/A285021(k) = -sqrt(5/6).

%t Numerator[Table[(-1/5)^n*Sqrt[Pi]/(Gamma[1/2-n]*Gamma[1+n]),{n,0,30}]]

%Y Cf. A285021 (denominators).

%K nonn,frac

%O 0,3

%A _Ralf Steiner_, Apr 08 2017