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Numerator of zeta({6}_n)/Pi^(6n).
1

%I #23 Jun 02 2022 08:31:31

%S 1,1,4,2,4,1,4,4,4,4,16,2,4,2,8,8,4,4,16,8,16,1,4,4,4,4,16,4,8,4,16,

%T 16,4,4,16,8,16,4,16,16,16,16,64,2,4,2,8,8,4,4,16,8,16,2,8,8,8,8,32,8,

%U 16,8,32,32,4,4,16,8,16,4,16

%N Numerator of zeta({6}_n)/Pi^(6n).

%C ({6}_n) is standard notation for multiple zeta values. It represents (6, ..., 6) where the multiplicity of 6 is n.

%H J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, <a href="https://arxiv.org/abs/hep-th/9611004">Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k</a>, arXiv:hep-th/9611004, 1996.

%H Roudy El Haddad, <a href="https://arxiv.org/abs/2102.00821">Multiple Sums and Partition Identities</a>, arXiv:2102.00821 [math.CO], 2021.

%H Roudy El Haddad, <a href="https://doi.org/10.7546/nntdm.2022.28.2.200-233">A generalization of multiple zeta value. Part 2: Multiple sums</a>. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.

%F a(n) = numerator(6*2^(6*n)/(6*n + 3)!).

%F a(n) = 2^(A000120(3*n + 1) - 1).

%F a(n) = 2^A240883(n).

%t a[n_] := Numerator[6*2^(6*n)/(6*n + 3)!]; Array[a, 71, 0]

%o (PARI) a(n) = 1 << (hammingweight(3*n+1) - 1);

%Y Cf. A351806 (denominators).

%Y Cf. A000120, A240883, A046988, A013664, A103345.

%K nonn,easy,frac

%O 0,3

%A _Roudy El Haddad_, Feb 22 2022