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A113788 Number of irreducible multiple zeta values at weight n.

%I A113788 #87 Jul 22 2020 05:24:21
%S A113788 0,1,1,0,1,0,1,1,1,1,2,2,3,3,4,5,7,8,11,13,17,21,28,34,45,56,73,92,
%T A113788 120,151,197,250,324,414,537,687,892,1145,1484,1911,2479,3196,4148,
%U A113788 5359,6954,9000,11687,15140,19672,25516,33166,43065,56010,72784,94716,123185
%N A113788 Number of irreducible multiple zeta values at weight n.
%C A113788 n * a(n) is the Möbius transform of the Perrin sequence A001608.
%C A113788 Number of unlabeled (i.e., defined up to a rotation) maximal independent sets of the n-cycle graph having n isomorphic representatives. - Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007
%H A113788 Danny Rorabaugh, <a href="/A113788/b113788.txt">Table of n, a(n) for n = 1..8000</a>
%H A113788 Kam Cheong Au, <a href="https://arxiv.org/abs/2007.03957">Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series</a>, arXiv:2007.03957 [math.NT], 2020. See 1st line of Table 1 (p. 6).
%H A113788 R. Bisdorff and J.-L. Marichal, <a href="https://arxiv.org/abs/math/0701647">Counting non-isomorphic maximal independent sets of the n-cycle graph</a>, arXiv:0701647 [math.CO], 2007-2008.
%H A113788 R. Bisdorff and J.-L. Marichal, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Marichal/marichal.html">Counting non-isomorphic maximal independent sets of the n-cycle graph</a>, JIS 11 (2008), #08.5.7.
%H A113788 D. J. Broadhurst and D. Kreimer, <a href="http://arXiv.org/abs/hep-th/9609128">Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops</a>, UTAS-PHYS-96-44; arXiv:hep-th/9609128, 1996.
%H A113788 D. J. Broadhurst and D. Kreimer, <a href="http://dx.doi.org/10.1016/S0370-2693(96)01623-1">Associated multiple zeta values with positive knots via Feynman diagrams up to 9 knots</a>, Phys. Lett B, 393 (1997), 403-412.
%H A113788 M. Waldschmidt, <a href="http://www.math.jussieu.fr/~miw/articles/pdf/MZV2011IMSc.pdf">Lectures on Multiple Zeta Values</a>, IMSC 2011.
%F A113788 a(n) = (1/n) * Sum_{d|n} mu(n/d)*Perrin(d), where Perrin(d) = A001608 starting with 0, 2, 3, ... .
%F A113788 a(n) = Sum_{d|n} mu(n/d)*A127687(d) = (1/n) * Sum_{d|n} mu(n/d)*A001608(d). - Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007
%F A113788 For p an odd prime, a(p) = Sum_{i=0..floor((p-3)/6)} (A(i)+B(i)-1)!/(A(i)!*B(i)!), where A(i) = (p-3)/2 - 3*i, and B(i) = 1 + 2*i. - _Richard Turk_, Sep 08 2015
%F A113788 a(n) ~ A060006^n / n. - _Vaclav Kotesovec_, Oct 09 2019
%p A113788 A113788 := proc(n::integer)
%p A113788     local resul,d;
%p A113788     resul :=0;
%p A113788     for d from 1 to n do
%p A113788         if n mod d = 0 then
%p A113788             resul := resul +numtheory[mobius](n/d)*A001608(d);
%p A113788         fi;
%p A113788     od:
%p A113788     RETURN(resul/n);
%p A113788 end: # _R. J. Mathar_, Apr 25 2006
%t A113788 (* p = A001608 *) p[n_] := p[n] = p[n-2] + p[n-3]; p[0] = 3; p[1] = 0; p[2] = 2; a[n_] := (1/n)*Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 56}] (* _Jean-François Alcover_, Jul 16 2012, from first formula *)
%o A113788 (Sage)
%o A113788 z = PowerSeriesRing(ZZ, 'z').gen().O(30)
%o A113788 r = (1 - (z**2 + z**3))
%o A113788 F = -z*r.derivative()/r
%o A113788 [sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # _F. Chapoton_, Apr 24 2020
%Y A113788 Cf. A001608, A127687, A125951.
%K A113788 nonn,easy
%O A113788 1,11
%A A113788 _R. J. Mathar_, Jan 27 2006