OFFSET
0,2
COMMENTS
Denominators are given by A124398.
The alternating sums over central binomial coefficients scaled by powers of 5, r(n) = Sum_{k=0..n} (-1)^k*binomial(2*k,k)/5^k, have the limit s = lim_{n-> infinity} r(n) = sqrt(5)/3. From the expansion of 1/sqrt(1+x) for x=4/5.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Rationals and more.
FORMULA
EXAMPLE
a(3)=17 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = a(3)/A124398(3).
MAPLE
seq(numer(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019
MATHEMATICA
Table[Numerator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k, 0, n}]], {n, 0, 20}] (* G. C. Greubel, Dec 25 2019 *)
PROG
(PARI) a(n) = numerator(sum(k=0, n, ((-1)^k)*binomial(2*k, k)/5^k)); \\ Michel Marcus, Aug 11 2019
(Magma) [Numerator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019
(Sage) [numerator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019
(GAP) List([0..20], n-> NumeratorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Nov 10 2006
STATUS
approved