A130545 Numerators of 2*Sum_{k=1..n} 1/binomial(2*k,k), n >= 1.

1, 4, 43, 307, 463, 10201, 24121, 88453, 1503743, 28571327, 680271, 54761843, 156462429, 111170677, 245020174253, 7595625419003, 2531875141141, 17723125990639, 655755661678837, 655755661685297, 867289746102097

Offset

1,2

Comments

Partial sums (in lowest terms) for a series of (2/27)*(9+2*Pi*sqrt(3)).

The rationals r(n) = 2*Sum_{k=1..n} 1/binomial(2*k,k) tend, in the limit n->infinity, to (2/27)*(9 + 2*Pi*sqrt(3)), which is approximately 1.472799718.

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise (with a misprint).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

W. Lang, Rationals and limit.

Formula

a(n) = numerator(r(n)), n >= 1, with the rationals defined above.

Example

Rationals r(n): 1, 4/3, 43/30, 307/210, 463/315, 10201/6930, 24121/16380, ....

Mathematica

Numerator[Table[2*Sum[1/Binomial[2k, k], {k, n}], {n, 30}]] (* Harvey P. Dale, Jul 30 2015 *)

Crossrefs

Denominators are given by A130546.

Cf. A130547/A130548 for s(n):=r(n)-2/3.

Sequence in context: A297649 A186678 A186768 * A027311 A198205 A277639

Adjacent sequences: A130542 A130543 A130544 * A130546 A130547 A130548

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Jul 13 2007

Status

approved