A285020 Numerator of binomial(2*n,n)/20^n.

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

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).

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: A194583 A346784 A060487 * A165781 A152095 A096797

Adjacent sequences: A285017 A285018 A285019 * A285021 A285022 A285023

Keyword

nonn,frac

Author

Ralf Steiner, Apr 08 2017

Status

approved