A050385 Reversion of Moebius function A008683.

1, 1, 3, 10, 39, 160, 691, 3081, 14095, 65757, 311695, 1496833, 7266979, 35608419, 175875537, 874698246, 4376646808, 22016578909, 111282845162, 564886771380, 2878498888625, 14719219809915, 75505990358779, 388451973679785

Offset: 1

Author

Christian G. Bower, Nov 15 1999

Keywords

• nonn
• nice

Comments

From David W. Wilson, May 17 2017: (Start)

Appears to equal the number of ways to partition Z into n residue classes. For example, a(4) = 10 since we can partition Z into 4 residue classes in 10 ways:

Z = ∪ {0 (mod 2), 1 (mod 4), 3 (mod 8), 7 (mod 8)}

Z = ∪ {0 (mod 2), 3 (mod 4), 1 (mod 8), 5 (mod 8)}

Z = ∪ {0 (mod 2), 1 (mod 6), 3 (mod 6), 5 (mod 6)}

Z = ∪ {1 (mod 2), 0 (mod 4), 2 (mod 8), 6 (mod 8)}

Z = ∪ {1 (mod 2), 2 (mod 4), 0 (mod 8), 4 (mod 8)}

Z = ∪ {1 (mod 2), 0 (mod 6), 2 (mod 6), 4 (mod 6)}

Z = ∪ {0 (mod 3), 1 (mod 3), 2 (mod 6), 5 (mod 6)}

Z = ∪ {0 (mod 3), 2 (mod 3), 1 (mod 6), 4 (mod 6)}

Z = ∪ {1 (mod 3), 2 (mod 3), 0 (mod 6), 3 (mod 6)}

Z = ∪ {0 (mod 4), 1 (mod 4), 2 (mod 4), 3 (mod 4)}

(End)

Unfortunately this conjecture is not correct; it fails at a(13). - Jeffrey Shallit, Nov 19 2017

The correct statement is that a(n) is the number of "natural exact covering systems" of cardinality n. These are covering systems (like the ones in David W. Wilson's comment) that are obtained by starting with the size-1 system x == 0 (mod 1) and successively choosing a congruence and "splitting" it into r >= 2 new congruences. - Jeffrey Shallit, Dec 07 2017

Links

• David W. Wilson, Table of n, a(n) for n = 1..1000
• Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
• I. P. Goulden, Andrew Granville, L. Bruce Richmond, and J. Shallit, Natural exact covering systems and the reversion of the Möbius series, Ramanujan J. (2019) Vol. 50, 211-235.
• I. P. Goulden, L. B. Richmond, and J. Shallit, Natural exact covering systems and the reversion of the Möbius series, arXiv:1711.04109 [math.NT], 2017-2018.
• N. J. A. Sloane, Transforms
• Index entries for reversions of series

Crossrefs

Cf. A008683, A050386, A294672, A295162.

Programs

Mathematica

InverseSeries[Sum[MoebiusMu[n] x^n, {n, 0, 25}] + O[x]^25] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Sep 29 2018 *)

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} mu(k) * A(x)^k. - Ilya Gutkovskiy, Apr 22 2020