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%I #14 Aug 30 2019 04:05:29
%S 1,5,47,949,33287,14273,7694047,400101469,1200312247,20405339951,
%T 4264717637359,328055232193,1275150714976991,1275150721602467,
%U 2125251205342781,246529139894912671,129920856734238187217,2122119257040297503,22216466502052353380347,164401852115363364287267
%N a(n) = numerator(Sum_{k=0..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.Numerators of partial sums of a certain series of inverse central binomial coefficients".
%C See A145565 for the denominators.
%C The limit of the rational partial sums r(n), defined below, for n->infinity is (4*sqrt(3)- Pi)*Pi/9. This limit is approximately 1.321776442.
%H W. Lang, <a href="/A145564/a145564.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, fifth identity.
%F a(n) = numerator(r(n)) with r(n)=sum(1/(binomial(2*k,k)*(k+1)),k=0..n), rationals in lowest terms.
%e Rationals r(n) (in lowest terms): [1, 5/4, 47/36, 949/720, 33287/25200, 14273/10800, 7694047/5821200,...].
%o (PARI) a(n) = numerator(sum(k=0, n, 1/(binomial(2*k,k)*(k+1)))); \\ _Michel Marcus_, Nov 08 2015
%K nonn,frac,easy
%O 0,2
%A _Wolfdieter Lang_, Oct 17 2008
%E New name based on formula by _Michel Marcus_, Nov 08 2015
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