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 A113788 Number of irreducible multiple zeta values at weight n. 8
 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, 120, 151, 197, 250, 324, 414, 537, 687, 892, 1145, 1484, 1911, 2479, 3196, 4148, 5359, 6954, 9000, 11687, 15140, 19672, 25516, 33166, 43065, 56010, 72784, 94716, 123185 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS n * a(n) is the Möbius transform of the Perrin sequence A001608. 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 LINKS Danny Rorabaugh, Table of n, a(n) for n = 1..8000 Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 1st line of Table 1 (p. 6). R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:0701647 [math.CO], 2007-2008. R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, JIS 11 (2008), #08.5.7. D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, UTAS-PHYS-96-44; arXiv:hep-th/9609128, 1996. D. J. Broadhurst and D. Kreimer, Associated multiple zeta values with positive knots via Feynman diagrams up to 9 knots, Phys. Lett B, 393 (1997), 403-412. M. Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011. FORMULA a(n) = (1/n) * Sum_{d|n} mu(n/d)*Perrin(d), where Perrin(d) = A001608 starting with 0, 2, 3, ... . 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 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 a(n) ~ A060006^n / n. - Vaclav Kotesovec, Oct 09 2019 MAPLE A113788 := proc(n::integer) local resul, d; resul :=0; for d from 1 to n do if n mod d = 0 then resul := resul +numtheory[mobius](n/d)*A001608(d); fi; od: RETURN(resul/n); end: # R. J. Mathar, Apr 25 2006 MATHEMATICA (* 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 *) PROG (Sage) z = PowerSeriesRing(ZZ, 'z').gen().O(30) r = (1 - (z**2 + z**3)) F = -z*r.derivative()/r [sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # F. Chapoton, Apr 24 2020 CROSSREFS Cf. A001608, A127687, A125951. Sequence in context: A036823 A035575 A036816 * A018243 A127207 A173513 Adjacent sequences: A113785 A113786 A113787 * A113789 A113790 A113791 KEYWORD nonn,easy AUTHOR R. J. Mathar, Jan 27 2006 STATUS approved

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Last modified November 28 05:31 EST 2022. Contains 358407 sequences. (Running on oeis4.)