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%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
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