login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 12 18:03 EDT 2020. Contains 335666 sequences. (Running on oeis4.)