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Numerators of partial sums of a certain alternating series of inverse central binomial coefficients.
6

%I #27 Aug 28 2019 17:42:03

%S 1,5,13,361,31,1193,31021,34467,5273479,1821745,220211,230450795,

%T 2880634987,1502939987,5896829249,12430516053889,1381168450513,

%U 3271188435379,2299645470079393,459929094015491,819873602375609,810854992749436603,311867304903633289

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

%C See A145558 for the denominators divided by 2.

%C 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.

%H Robert Israel, <a href="/A145557/b145557.txt">Table of n, a(n) for n = 1..118</a>

%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. 12, p. 39.

%H M. L. Glasser, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Glasser/glasser2.html">A Generalized Apery Series</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.4.3.

%H W. Lang, <a href="/A145557/a145557.txt">Rationals and more.</a>

%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.

%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.

%F 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 Rationals r(n) (in lowest terms): [1/2, 5/12, 13/30, 361/840, 31/72, 1193/2772, 31021/72072,...].

%p R:= 0;

%p for n from 1 to 100 do

%p R:= R + (-1)^(n+1)/(n*binomial(2*n,n));

%p a[n]:=numer(R);

%p od:

%p seq(a[i],i=1..100); # _Robert Israel_, Jun 16 2014

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

%Y A145375/A145556.

%K nonn,frac,easy

%O 1,2

%A _Wolfdieter Lang_, Oct 17 2008