A226226 Number of alignments of n points with no singleton cycles

1, 0, 1, 2, 12, 64, 470, 3828, 36456, 387840, 4603392, 60061440, 855664656, 13207470912, 219609303888, 3912940891104, 74377769483520, 1502277409668096, 32130095812624512, 725400731911792896, 17240044059713320704, 430231117562438446080, 11248105572520779755520

Offset

0,4

Comments

a(n) is the number of labeled sequences of cycles, where no cycle has size 1.

References

P. Flajolet and R. Segdewick, Analytic Combinatorics, Cambridge University Press, 2009, page 119

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

E.g.f.: 1/(1+x-log(1/(1-x)))

a(n) ~ n!*c/(1-c)^(n+2), where c = -LambertW(-exp(-2)) = 0.158594339563... - Vaclav Kotesovec, Jun 02 2013

a(0) = 1; a(n) = Sum_{k=0..n-2} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

Example

For n=4, the a(4)=12 alignments with no singletons are: 1234, 1243, 1324, 1342, 1423, 1432, 12|34, 13|24, 14|23, 23|14, 24|13, 34|12.

Mathematica

Range[0, 50]! CoefficientList[ Series[(1 + z - Log[1/(1 - z)])^(-1), {z, 0, 50}], z]

Prog

(PARI) x='x+O('x^66); Vec(serlaplace(1/(1+x-log(1/(1-x))))) \\ Joerg Arndt, Jun 01 2013

Crossrefs

The alignments with singletons included are given by A007840.

Sequence in context: A025599 A162973 A321989 * A126737 A345697 A097632

Adjacent sequences: A226223 A226224 A226225 * A226227 A226228 A226229

Keyword

nonn

Author

Ricardo Bittencourt, May 31 2013

Status

approved