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A226226
Number of alignments of n points with no singleton cycles
9
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
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
KEYWORD
nonn
AUTHOR
Ricardo Bittencourt, May 31 2013
STATUS
approved