A032181 Number of ways to partition n labeled elements into pie slices each of at least 2 elements.

0, 1, 1, 4, 11, 56, 267, 1730, 11643, 93532, 804563, 7789838, 81109107, 923080328, 11250876171, 147433014058, 2056359699659, 30514968348500, 479058943716579, 7942752339181286, 138576460230586755, 2539126631897727808, 48733588613803358939

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Links

• C. G. Bower, Transforms (2).
• S. Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938 [math.CO], 2013-2015.
• Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
• Index entries for sequences related to necklaces.

Programs

Mathematica

Rest[CoefficientList[Series[Log[1/(2+x-E^x)], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 30 2013 *)

Formula

"CIJ" (necklace, indistinct, labeled) transform of 0, 1, 1, 1...

E.g.f.: A(x) = log(1/(2+x-exp(x))).

a(n) = n! * sum(k=1..n, sum(j=0..k, binomial(k,j) *stirling2(n-k+j,j) *j!/(n-k+j)! *(-1)^(k-j))/k). - Vladimir Kruchinin, Feb 01 2011

a(n) ~ (n-1)! / (-LambertW(-1,-exp(-2))-2)^n. - Vaclav Kotesovec, Sep 30 2013