%I A145557
%S A145557 1,5,13,361,31,1193,31021,34467,5273479,1821745,220211,230450795,2880634987,
%T A145557 1502939987,5896829249,12430516053889,1381168450513,3271188435379,2299645470079393,
%U A145557 459929094015491,819873602375609,810854992749436603,311867304903633289
%N A145557 Numerators of partial sums of a certain alternating series of inverse 
               central binomial coefficients.
%C A145557 See A145558 for the denominators divided by 2.
%C A145557 The limit of the rational partial sums r(n), defined below, for n->infinity 
               is 2*(2*phi-1)*ln(phi)/5, with phi:=(1+sqrt(5))/2 (golden section). 
               This limit is approximately 0.4304089412.
%D A145557 C. Elsner, On recurrence formulae for sums involving binomial coefficients, 
               Fib. Q., 43,1 (2005), 31-45. Eq.12, p.39.
%D A145557 A. J. van der Poorten, Some wonderful formulae...Footnote to Apery's 
               proof of the irrationality of zeta(3), S\'eminaire Delange-Pisot-Poitou. 
               Th\'eorie des nombres, tome 20, no 2 (1978-1979), exp, no 29, p.1-7. 
               p. 29-02 Available via http://www.numdam.org/numdam-bin/qrech
%D A145557 R. Sprugnoli, Sums of reciprocals of the central binomial coefficients, 
               Integers: electronic journal of combinatorial number theory, 6 (2006) 
               #A27, 1-18.
%H A145557 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A145557.text">
               Rationals and more.</a>
%F A145557 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).
%e A145557 Rationals r(n) (in lowest terms): [1/2, 5/12, 13/30, 361/840, 31/72, 
               1193/2772, 31021/72072,...].
%Y A145557 A145375/A145556.
%Y A145557 Sequence in context: A085554 A067135 A122900 this_sequence A012033 A007540 
               A157250
%Y A145557 Adjacent sequences: A145554 A145555 A145556 this_sequence A145558 A145559 
               A145560
%K A145557 nonn,frac,easy
%O A145557 1,2
%A A145557 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 17 2008