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A115750
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a(n) = Sum_{k=0..2*n} binomial(2*n, k)*binomial(3*n, 2*n + 2*k).
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1
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1, 3, 19, 138, 1051, 8228, 65602, 529840, 4320507, 35492475, 293285544, 2435133110, 20299183738, 169780446228, 1424093337728, 11974638998288, 100907444665595, 851939678134229, 7204872937244995, 61023558185533392
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OFFSET
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0,2
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COMMENTS
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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]
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REFERENCES
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Stéphane Fischler and Tanguy Rivoal, Un exposant de densité en approximation rationnelle, preprint, 2006.
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=sum(k=0, 2*n, binomial(2*n, k)*binomial(3*n, 2*n+2*k))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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