A130416 Numerator of partial sums for a series of (17/18)*Zeta(4) = (17/1680)*Pi^4.

1, 49, 6623, 741857, 13247611, 3060203141, 13645449045719, 218327192834879, 100212182125865461, 1904031462407822767, 2534265876944902342877, 58288115171766608401171, 128058989033214718801833487

Offset: 1

Author

Wolfdieter Lang, Jul 13 2007

Keywords

• nonn
• frac
• easy

Comments

Denominators are given by A130417.

The rationals r(n) = 2*Sum_{k=1..n} 1/(k^4*binomial(2*k,k)) tend, in the limit n->infinity, to (18/17)*Zeta(4) = (17/1680)*Pi^4, approximately 1.022194166.

Examples

			Rationals: 1, 49/48, 6623/6480, 741857/725760, 13247611/12960000, ...
		

References

• L. Berggren, T. Borwein and P. Borwein, Pi: A Source Book, Springer, New York, 1997, p. 687.
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.

Links

• W. Lang, Rationals and limit.
• A. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report, Math. Intelligencer 1 (1978/79), no. 4, 195-203; reprinted in Pi: A Source Book, pp. 439-447, footnote 10, p. 446 (conjecture).

Crossrefs

Partial sums for Zeta(4): A007410/A007480.

Formula

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