OFFSET
1,2
COMMENTS
See A145558 for the denominators divided by 2.
The limit of the rational partial sums r(n), defined below, for n->infinity is 2*(2*phi-1)*log(phi)/5, with phi:=(1+sqrt(5))/2 (golden section). This limit is approximately 0.4304089412.
LINKS
Robert Israel, Table of n, a(n) for n = 1..118
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. See Eq. 12, p. 39.
M. L. Glasser, A Generalized Apery Series, Journal of Integer Sequences, Vol. 15 (2012), #12.4.3.
W. Lang, Rationals and more.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18.
A. J. van der Poorten, Some wonderful formulas...Footnote to Apery's proof of the irrationality of zeta(3), Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 20, no. 2 (1978-1979), exp, no. 29, pp. 1-7, pp. 29-02.
FORMULA
a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n):=sum(((-1)^(k+1))/(k*binomial(2*k,k)),k=1..n).
EXAMPLE
Rationals r(n) (in lowest terms): [1/2, 5/12, 13/30, 361/840, 31/72, 1193/2772, 31021/72072,...].
MAPLE
R:= 0;
for n from 1 to 100 do
R:= R + (-1)^(n+1)/(n*binomial(2*n, n));
a[n]:=numer(R);
od:
seq(a[i], i=1..100); # Robert Israel, Jun 16 2014
PROG
(PARI) vector(50, n, numerator(sum(k=1, n, (-1)^(k+1)/(k*binomial(2*k, k))))) \\ Michel Marcus, Oct 13 2014
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Oct 17 2008
STATUS
approved