OFFSET
1,2
COMMENTS
For the corresponding numerator sequence see A119951.
The series s:=Sum_{k>=1} C(k)/2^(2*(k-1)), with C(n):=A000108(n) (Catalan numbers) converges by Raabe's test. The value for s is 4 (see A119951).
Asymptotically, C(n)/2^(2*(k-1)) ~ 4/(sqrt(Pi)*k^(3/2)) (see Mathworld). The sum of the asymptotic values from k = 1 to infinity is (4/sqrt(Pi))*Zeta(3/2) = 5.895499840 (Maple10, 10 digits).
The partial sums r(n):=Sum_{k=1..n} C(k)/2^(2*(k-1)) are rationals (written in lowest terms).
For the rationals r(n) see the W. Lang link under A119951.
a(n) appears to be the denominator of Catalan(n)/4^(n-1) but I have no proof of this. - Groux Roland, Dec 11 2010
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = denominator(r(n)) with the rationals r(n) defined above.
MATHEMATICA
Denominator[Table[Sum[CatalanNumber[k]/2^(2*(k - 1)), {k, 1, n}], {n, 1, 50}]] (* G. C. Greubel, Feb 08 2017 *)
PROG
(PARI) for(n=1, 50, print1(denominator(sum(k=1, n, binomial(2*k, k)/((k+1)*2^(2*k-2)))), ", ")) \\ G. C. Greubel, Feb 08 2017
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Wolfdieter Lang, Jul 20 2006
EXTENSIONS
First comment corrected by Harvey P. Dale, Oct 09 2017
STATUS
approved