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%I #29 Jun 09 2020 14:37:19
%S 1,3,19,138,1051,8228,65602,529840,4320507,35492475,293285544,
%T 2435133110,20299183738,169780446228,1424093337728,11974638998288,
%U 100907444665595,851939678134229,7204872937244995,61023558185533392
%N a(n) = Sum_{k=0..2*n} binomial(2*n, k)*binomial(3*n, 2*n + 2*k).
%C Integer sequence arising in a Beuker's approximation to Pi (see prop. 8 page 23 of the reference). [It does not appear to be in the given reference. - _Petros Hadjicostas_, Jun 09 2020]
%D Stéphane Fischler and Tanguy Rivoal, Un exposant de densité en approximation rationnelle, preprint, 2006.
%H Vaclav Kotesovec, <a href="/A115750/b115750.txt">Table of n, a(n) for n = 0..1000</a>
%H Stéphane Fischler and Tanguy Rivoal, <a href="https://www.imo.universite-paris-saclay.fr/~fischler/exposant.pdf">Un exposant de densité en approximation rationnelle</a>, preprint, 2006.
%H Stéphane Fischler and Tanguy Rivoal, <a href="http://rivoal.perso.math.cnrs.fr/articles/exposant.pdf">Un exposant de densité en approximation rationnelle</a>, preprint, 2006.
%H Stéphane Fischler and Tanguy Rivoal, <a href="http://dx.doi.org/10.1155/IMRN/2006/95418">Un exposant de densité en approximation rationnelle</a>, Int. Math. Res. Notices, Vol. 2006 (2006), Article ID 95418, 48 pp.
%H Tanguy Rivoal, <a href="http://rivoal.perso.math.cnrs.fr/">Homepage</a>.
%F From _Vaclav Kotesovec_, Jun 07 2019: (Start)
%F a(n) ~ c * d^n / sqrt(Pi*n), where d = 27 * (1 - 2*r)^(-1 + 2*r) * (2 - r)^(-2 + r) / (2^(2*r) * r^r * (1+r)^(2*(1 + r))) = 8.6988890096304955678255243852749992..., r = 1/6 + (27*sqrt(139) - 5)^(1/3) / (6*2^(2/3)) - 37/(6*(54*sqrt(139) - 10)^(1/3)) = 0.1591594336002991371303884200119396931041597457946... is the real root of the equation -2 + 13*r - 4*r^2 + 8*r^3 = 0 and c = 0.670323490697444616208038892968942176908111537748186024028564941159... is the positive real root of the equation -12 + 328*c^2 - 2919*c^4 + 5004*c^6 = 0.
%F Equivalently, r is the root of the equation arctanh(1-r) = 2*arctanh((4*r+1)/3).
%F (End)
%F a(n) = binomial(3*n, 2*n)*hypergeometric([1/2-n/2, -2*n, -n/2], [n+1/2, n+1], -1). - _Peter Luschny_, Jun 09 2020
%t Table[Sum[Binomial[2*n,k]*Binomial[3*n,2*n+2*k], {k,0,2*n}], {n,0,20}] (* _Vaclav Kotesovec_, Jun 07 2019 *)
%o (PARI) a(n)=sum(k=0,2*n,binomial(2*n,k)*binomial(3*n,2*n+2*k))
%K nonn
%O 0,2
%A _Benoit Cloitre_, Mar 24 2006
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