A145566 - OEIS

%I #28 Jun 12 2022 07:06:35

%S 1,23,33,199,10957,11873,35621,4844519,2789277,2789279,705687707,

%T 1764219339,3175594841,26312071601,79968060793,479808364823,

%U 57097195415809,234732914489081,704198743468603,28872148482226289,17992788184577863,161935093661205289

%N a(n) = numerator(6 * Sum_{k=2..n} 1/(binomial(2*k, k)*(k-1))).

%C Previous name was: "Numerators of partial sums of a certain series of inverse central binomial coefficients".

%C See A145567 for the denominators/6.

%C The limit of the rational partial sums r(n), defined below, for n->infinity is (9 - sqrt(3)*Pi)/3. This limit is approximately 1.186200635.

%H Wolfdieter Lang, <a href="/A145566/a145566.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. Theorem 3.4, sixth identity times 6.

%e Rationals 6*r(n) (in lowest terms): [1, 23/20, 33/28, 199/168, 10957/9240, 11873/10010, 35621/30030, 4844519/4084080,...].

%p a := n -> numer(6*add(1/(binomial(2*k, k)*(k-1)), k=2..n)):

%p seq(a(n), n = 2..23); # _Peter Luschny_, Jun 12 2022

%o (PARI) a(n) = numerator(6*sum(k=2, n, 1/(binomial(2*k,k)*(k-1)))); \\ _Michel Marcus_, Nov 08 2015; with factor 6 by _Georg Fischer_, Jun 11 2022

%K nonn,frac,easy

%O 2,2

%A _Wolfdieter Lang_, Oct 17 2008

%E New name based on formula by _Michel Marcus_, Nov 08 2015

%E Definition amended by _Georg Fischer_, Jun 12 2022