A011796 Number of irreducible alternating Euler sums of depth 6 and weight 6+2n.

1, 3, 9, 20, 42, 75, 132, 212, 333, 497, 728, 1026, 1428, 1932, 2583, 3384, 4389, 5598, 7084, 8844, 10962, 13442, 16380, 19776, 23751, 28301, 33561, 39536, 46376, 54081, 62832, 72624, 83655, 95931, 109668, 124866, 141778, 160398

Offset: 1

Author

N. J. A. Sloane, David Broadhurst

Keywords

• nonn
• easy

Comments

a(n-6) is the number of aperiodic necklaces (Lyndon words) with 6 black beads and n-6 white beads.

References

• J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.

Links

• David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
• David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See pp. 3, 11, 19.
• Index entries for sequences related to Lyndon words
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,4,-3,-3,4,1,-1,-3,1,2,-1).

Crossrefs

Cf. A000031, A001037, A051168.

Programs

Maple

a:= n-> (Matrix([[42, 20, 9, 3, 1, 0$7, -1, -4, -9]]). Matrix(15, (i,j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -3, -1, 1, 4, -3, -3, 4, 1, -1, -3, 1, 2, -1][i] else 0 fi)^(n-5))[1,1]: seq(a(n), n=1..50); # Alois P. Heinz, Aug 04 2008

Mathematica

a[n_] := Sum[Binomial[(n+6)/d, 6/d]*MoebiusMu[d],{d, Divisors[GCD[6, n+6]]}]/(n+6); Array[a, 40] (* Jean-François Alcover, Feb 02 2015 *)

Formula

G.f.: x*(1+x+2*x^2+2*x^3+3*x^4+2*x^6+x^7)/((1-x)^2*(1-x^2)^2*(1-x^3)*(1-x^6)).

G.f.: (1/(1-x)^6-1/(1-x^2)^3-1/(1-x^3)^2+1/(1-x^6))/6. - Herbert Kociemba, Oct 23 2016

a(n) = T(n,6), array T as in A051168.