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
Keywords
Examples
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.
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
Crossrefs
The alignments with singletons included are given by A007840.
Programs
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Mathematica
Range[0, 50]! CoefficientList[ Series[(1 + z - Log[1/(1 - z)])^(-1), {z, 0, 50}], z] -
PARI
x='x+O('x^66); Vec(serlaplace(1/(1+x-log(1/(1-x))))) \\ Joerg Arndt, Jun 01 2013
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
Comments