A020999 Conjectured number of irreducible multiple zeta values of depth n and weight 3n (confirmed up to n=7).

1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 8, 10, 14, 17, 24, 30, 41, 53, 72, 93, 126, 165, 222, 293, 395, 522, 704, 936, 1259, 1681, 2263, 3027, 4079, 5470, 7371, 9906, 13361, 17980, 24271, 32712, 44182, 59626, 80598, 108879, 147285, 199165

Offset: 0

Author

David Broadhurst

Keywords

• nonn
• nice

Links

• D. J. Broadhurst, Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams
• R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, sequence gamma_{3,j}^(A).

Crossrefs

Cf. A014097.

Programs

Mathematica

max = 50; ClearAll[a]; coes = CoefficientList[ Series[ Product[ (1-x^n)^a[n-1], {n, 0, max}] - (1-x-x^4), {x, 0, max}] /. 0^A020999%20=%20Table%5B%20a%5Bn%5D,%20%7Bn,%200,%20max-1%7D%5D(*%20From%20_Jean-Fran%C3%A7ois%20Alcover"> -> 1, x]; eq = Rest[ Thread[ coes == 0]]; s[1] = Solve[ eq[[1]], a[0]][[1]]; a[0] = a[0] /. s[1][[1]]; Print[a[0]]; Do[ eq = Rest[eq] /. s[n]; s[n+1] = Solve[ eq[[1]], a[n]][[1]]; a[n] = a[n] /. s[n+1][[1]]; Print[a[n]], {n, 1, max-1}]; A020999 = Table[ a[n], {n, 0, max-1}](* From _Jean-François Alcover, Jan 31 2012, after formula *)

Formula

Product_n (1-x^n)^{a(n)} = 1-x-x^4; equivalently, a(n) = (1/n) sum_{ d divides n } mu(n/d) A014097(d).