A113788 Number of irreducible multiple zeta values at weight n.

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

Offset: 1

Author

R. J. Mathar, Jan 27 2006

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A001608, A127687, A125951.

Programs

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 *)

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

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