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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Mar 24 2006
STATUS
approved