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 A115750 a(n) = Sum_{k=0..2*n} binomial(2*n, k)*binomial(3*n, 2*n + 2*k). 1
 1, 3, 19, 138, 1051, 8228, 65602, 529840, 4320507, 35492475, 293285544, 2435133110, 20299183738, 169780446228, 1424093337728, 11974638998288, 100907444665595, 851939678134229, 7204872937244995, 61023558185533392 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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] REFERENCES Stéphane Fischler and Tanguy Rivoal, Un exposant de densité en approximation rationnelle, preprint, 2006. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 Stéphane Fischler and Tanguy Rivoal, Un exposant de densité en approximation rationnelle, preprint, 2006. Stéphane Fischler and Tanguy Rivoal, Un exposant de densité en approximation rationnelle, preprint, 2006. Stéphane Fischler and Tanguy Rivoal, Un exposant de densité en approximation rationnelle, Int. Math. Res. Notices, Vol. 2006 (2006), Article ID 95418, 48 pp. Tanguy Rivoal, Homepage. FORMULA From Vaclav Kotesovec, Jun 07 2019: (Start) 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. Equivalently, r is the root of the equation arctanh(1-r) = 2*arctanh((4*r+1)/3). (End) 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 MATHEMATICA 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 *) PROG (PARI) a(n)=sum(k=0, 2*n, binomial(2*n, k)*binomial(3*n, 2*n+2*k)) CROSSREFS Sequence in context: A094661 A094662 A321349 * A156894 A221374 A073515 Adjacent sequences:  A115747 A115748 A115749 * A115751 A115752 A115753 KEYWORD nonn AUTHOR Benoit Cloitre, Mar 24 2006 STATUS approved

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Last modified August 8 09:40 EDT 2022. Contains 356009 sequences. (Running on oeis4.)