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 A285020 Numerator of binomial(2*n,n)/20^n. 2
 1, 1, 3, 1, 7, 63, 231, 429, 1287, 2431, 46189, 88179, 676039, 52003, 200583, 1938969, 60108039, 116680311, 90751353, 176726319, 6892326441, 13456446861, 52602474093, 20583576819, 322476036831, 15801325804719, 61989816618513, 121683714103007, 191217265019011, 375840831244263, 7391536347803839 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) is for p=1, q=5. Generally for p,q in N, p>0, q>1: Sum_{k>=0}(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=sqrt(q/(q-p)). Sum_{k>=0}(-1)^k*(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=sqrt(q/(q+p)). Sum_{k>=0}(-1)^(k+1)*(-p/q)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))=-sqrt(q/(q+p)). a(n) is the numerator of binomial(2*n,n)/20^n. - Robert Israel, Apr 09 2017 LINKS Chai Wah Wu, Table of n, a(n) for n = 0..1000 FORMULA a(n)/A285021(n) = (-1/5)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} a(k)/A285021(k) = sqrt(5)/2. Sum_{k>=0} (-1)^k*a(k)/A285021(k) = sqrt(5/6). Sum_{k>=0} (-1)^(k+1)*a(k)/A285021(k) = -sqrt(5/6). MAPLE P:=proc(q)  numer((-1/5)^q*sqrt(Pi)/(GAMMA(1/2-q)*GAMMA(1+q))); end: seq(P(i), i=0..20); # Paolo P. Lava, Apr 10 2017 MATHEMATICA Numerator[Table[(-1/5)^n*Sqrt[Pi]/(Gamma[1/2-n]*Gamma[1+n]), {n, 0, 30}]] CROSSREFS Cf. A285021 (denominators). Sequence in context: A282685 A194583 A060487 * A165781 A152095 A096797 Adjacent sequences:  A285017 A285018 A285019 * A285021 A285022 A285023 KEYWORD nonn,frac AUTHOR Ralf Steiner, Apr 08 2017 STATUS approved

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Last modified July 29 07:28 EDT 2021. Contains 346340 sequences. (Running on oeis4.)