OFFSET
1,2
COMMENTS
For the corresponding denominator sequence see A120069.
The asymptotics for C(n)/2^(2*(k-1)) is 4/(sqrt(Pi)*k^(3/2)) (see the E. Weisstein link, also for references). The sum over the asymptotic values from k=1..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).
The above partial sums are equal to 4 - binomial(2n+2,n+1)/2^(2n-1). - Pieter Mostert, Oct 12 2012
The series s = Sum_{k>=1} C(k)/2^(2*(k-1)), with C(n):=A000108(n) (Catalan numbers), converges by J. L. Raabe's criterion. See the Meschkowski reference for Raabe's criterion and the example given there. The series he gives as an example can be rewritten as (1 + 4*s)/2. From the expansion of sqrt(1+x) for |x|<=1 one finds for x=-1 the value s=4 (see the W. Lang link).
This sequence was essential for unraveling the structure of the row sums A160466 of the Eta triangle A160464. - Johannes W. Meijer, May 24 2009
REFERENCES
H. Meschkowski, Unendliche Reihen, 2., verb. u. erw. Aufl., Mannheim, Bibliogr. Inst., 1982, p. 32.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, Rationals r(n) and more.
Eric Weisstein's World of Mathematics, Catalan numbers, see eq.(10).
FORMULA
a(n) = numerator of Sum_{k=1..n} C(k)/2^(2*(k-1)).
a(n-1) = numerator of (1/4^n)*Sum_{i=0..n-1} (binomial(2*(i+1), i+1)*binomial(2*(n-i), n-i)), for n>=1. - Johannes W. Meijer, May 24 2009
a(n) = (2^n-(2*n+2)!/(2^(n+1)*(n+1)!^2))*gcd((n+1)!,2^(n+1)). - Gary Detlefs, Nov 06 2020
EXAMPLE
Rationals r(n): [1, 3/2, 29/16, 65/32, 281/128, 595/256, 9949/4096, 20613/8192, ...]
MATHEMATICA
Numerator[Table[(1/4^n)*Sum[Binomial[2*(i + 1), i + 1]*Binomial[2*(n - i), n - i], {i, 0, n - 1}], {n, 1, 50}]] (* G. C. Greubel, Jan 31 2017 *)
PROG
(PARI) for(n=1, 25, print1(numerator(sum(i=0, n-1, binomial(2*(i+1), i+1)* binomial(2*(n-i), n-i))/4^n), ", ")) \\ G. C. Greubel, Jan 31 2017
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Wolfdieter Lang, Jul 20 2006
STATUS
approved