A145375 - OEIS

%I #15 Aug 29 2019 17:16:23

%S 1,1,23,31,47,1031,26827,134107,455989,8663665,4331849,187279,

%T 4981622687,747243353,173360460899,1074834852769,233659750871,

%U 926581770421,198844447947463,6856705101503,1630524473145553,350562761725846217,97378544923877951,42247307182355837

%N Numerators of partial sums of the alternating series of inverse central binomial coefficients.

%C See A145556 for the denominators.

%C The limit of the rational partial sums r(n), defined below, for n->infinity is (1 + 4*log(phi)/(2*phi-1))/5, with phi:=(1+sqrt(5))/2 (golden section). This limit is approximately 0.3721635764.

%H C. Elsner, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-5.pdf">On recurrence formulas for sums involving binomial coefficients</a>, Fib. Q., 43,1 (2005), 31-45. See Eq. 13, p. 39.

%H W. Lang, <a href="/A145375/a145375.txt">Rationals and more.</a>

%H A. J. van der Poorten, <a href="http://www.numdam.org/item?id=SDPP_1978-1979__20_2_A6_0">Some wonderful formulas...Footnote to Apery's proof of the irrationality of zeta(3)</a>, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 20, no. 2 (1978-1979), exp, no. 29, pp. 1-7, pp. 29-02.

%H Renzo Sprugnoli, <a href="http://www.emis.de/journals/INTEGERS/papers/g27/g27.Abstract.html">Sums of reciprocals of the central binomial coefficients</a>, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18.

%F a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n):=sum(((-1)^(k+1))/binomial(2*k,k),k=1..n).

%e Rationals r(n) (in lowest terms): [1/2, 1/3, 23/60, 31/84, 47/126, 1031/2772, 26827/72072, ...].

%o (PARI) vector(50, n, numerator(sum(k=1, n,(-1)^(k+1)/binomial(2*k,k)))) \\ _Michel Marcus_, Oct 13 2014

%Y A145557/A145558.

%K nonn,frac,easy

%O 1,3

%A _Wolfdieter Lang_, Oct 17 2008, Nov 17 2008, Nov 25 2008