A130553 Numerators of partial sums for a series for 2*Pi*sqrt(3)/9.

1, 7, 6, 169, 1523, 133, 72623, 87149, 823077, 15638477, 46915441, 13834041, 224803169, 6936783521, 5587964507, 4157445593923, 12472336782289, 170187831339, 71785227258967, 153825486983593, 4905323862699739

Offset: 1

Author

Wolfdieter Lang, Jul 13 2007

Keywords

• nonn
• frac
• easy

Comments

Denominators are given in A130554.

The rational sequence r(n) = 2*Sum_{j=1..n} 1/(j*binomial(2*j,j)), n >= 1, tends, in the limit as n->infinity, to 2*Pi*sqrt(3)/9, which is approximately 1.209199577.

With offset 0 the rationals r(n) coincide with Sum_{j=0..n} 1/((2*j+1)*binomial(2*j,j)), n >= 0. See e.g. the Sprugnoli reference. [Wolfdieter Lang, Oct 17 2008]

Examples

			Rationals r(n): 1, 7/6, 6/5, 169/140, 1523/1260, 133/110, 72623/60060, ....
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.

Links

• C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. 2*Eq. 10, p. 38. [_Wolfdieter Lang_, Oct 17 2008]
• W. Lang, Rationals and limit.
• Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18. [_Wolfdieter Lang_, Oct 17 2008]

Programs

PARI

a(n) = numerator(2*sum(j=1, n, 1/(j*binomial(2*j,j)))); \\ Michel Marcus, Nov 08 2015

Formula

a(n) = numerator(r(n)), n >= 1, with the rationals r(n) defined above and taken in lowest terms.