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%I #14 May 09 2013 09:58:16
%S 1,0,0,1,1,1,1,1,2,2,3,3,5,6,8,10,14,17,24,30,41,53,72,93,126,165,222,
%T 293,395,522,704,936,1259,1681,2263,3027,4079,5470,7371,9906,13361,
%U 17980,24271,32712,44182,59626,80598,108879,147285,199165
%N Conjectured number of irreducible multiple zeta values of depth n and weight 3n (confirmed up to n=7).
%H D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9612012">Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, sequence gamma_{3,j}^(A).
%F Product_n (1-x^n)^{a(n)} = 1-x-x^4; equivalently, a(n) = (1/n) sum_{ d divides n } mu(n/d) A014097(d).
%t max = 50; ClearAll[a]; coes = CoefficientList[ Series[ Product[ (1-x^n)^a[n-1], {n, 0, max}] - (1-x-x^4), {x, 0, max}] /. 0^_ -> 1, x]; eq = Rest[ Thread[ coes == 0]]; s[1] = Solve[ eq[[1]], a[0]][[1]]; a[0] = a[0] /. s[1][[1]]; Print[a[0]]; Do[ eq = Rest[eq] /. s[n]; s[n+1] = Solve[ eq[[1]], a[n]][[1]]; a[n] = a[n] /. s[n+1][[1]]; Print[a[n]], {n, 1, max-1}]; A020999 = Table[ a[n], {n, 0, max-1}](* From _Jean-François Alcover_, Jan 31 2012, after formula *)
%Y Cf. A014097.
%K nonn,nice
%O 0,9
%A _David Broadhurst_
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