# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a285020 Showing 1-1 of 1 %I A285020 #32 Mar 11 2024 01:50:29 %S A285020 1,1,3,1,7,63,231,429,1287,2431,46189,88179,676039,52003,200583, %T A285020 1938969,60108039,116680311,90751353,176726319,6892326441,13456446861, %U A285020 52602474093,20583576819,322476036831,15801325804719,61989816618513,121683714103007,191217265019011,375840831244263,7391536347803839 %N A285020 Numerator of binomial(2*n,n)/20^n. %C A285020 a(n) is for p=1, q=5. Generally for p,q in N, p>0, q>1: %C A285020 Sum_{k>=0}(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=sqrt(q/(q-p)). %C A285020 Sum_{k>=0}(-1)^k*(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=sqrt(q/(q+p)). %C A285020 Sum_{k>=0}(-1)^(k+1)*(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=-sqrt(q/(q+p)). %C A285020 a(n) is the numerator of binomial(2*n,n)/20^n. - _Robert Israel_, Apr 09 2017 %H A285020 Chai Wah Wu, Table of n, a(n) for n = 0..1000 %F A285020 a(n)/A285021(n) = (-1/5)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). %F A285020 Sum_{k>=0} a(k)/A285021(k) = sqrt(5)/2. %F A285020 Sum_{k>=0} (-1)^k*a(k)/A285021(k) = sqrt(5/6). %F A285020 Sum_{k>=0} (-1)^(k+1)*a(k)/A285021(k) = -sqrt(5/6). %t A285020 Numerator[Table[(-1/5)^n*Sqrt[Pi]/(Gamma[1/2-n]*Gamma[1+n]),{n,0,30}]] %Y A285020 Cf. A285021 (denominators). %K A285020 nonn,frac %O A285020 0,3 %A A285020 _Ralf Steiner_, Apr 08 2017 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE