A130416 - OEIS

%I #12 Aug 30 2019 03:54:54

%S 1,49,6623,741857,13247611,3060203141,13645449045719,218327192834879,

%T 100212182125865461,1904031462407822767,2534265876944902342877,

%U 58288115171766608401171,128058989033214718801833487

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

%C Denominators are given by A130417.

%C 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.

%D L. Berggren, T. Borwein and P. Borwein, Pi: A Source Book, Springer, New York, 1997, p. 687.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.

%H W. Lang, <a href="/A130416/a130416.txt">Rationals and limit.</a>

%H A. van der Poorten, <a href="http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf"> A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report</a>, Math. Intelligencer 1 (1978/79), no. 4, 195-203; reprinted in Pi: A Source Book, pp. 439-447, footnote 10, p. 446 (conjecture).

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

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

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

%K nonn,frac,easy

%O 1,2

%A _Wolfdieter Lang_, Jul 13 2007