cp's OEIS Frontend

A226226 Number of alignments of n points with no singleton cycles.

%I A226226 #23 Jul 23 2025 05:26:32
%S A226226 1,0,1,2,12,64,470,3828,36456,387840,4603392,60061440,855664656,
%T A226226 13207470912,219609303888,3912940891104,74377769483520,
%U A226226 1502277409668096,32130095812624512,725400731911792896,17240044059713320704,430231117562438446080,11248105572520779755520
%N A226226 Number of alignments of n points with no singleton cycles.
%C A226226 a(n) is the number of labeled sequences of cycles, where no cycle has size 1.
%D A226226 P. Flajolet and R. Segdewick, Analytic Combinatorics, Cambridge University Press, 2009, page 119
%H A226226 Vincenzo Librandi, <a href="/A226226/b226226.txt">Table of n, a(n) for n = 0..200</a>
%F A226226 E.g.f.: 1/(1+x-log(1/(1-x)))
%F A226226 a(n) ~ n!*c/(1-c)^(n+2), where c = -LambertW(-exp(-2)) = 0.158594339563... - _Vaclav Kotesovec_, Jun 02 2013
%F A226226 a(0) = 1; a(n) = Sum_{k=0..n-2} binomial(n,k) * (n-k-1)! * a(k). - _Ilya Gutkovskiy_, Apr 26 2021
%e A226226 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.
%t A226226 Range[0, 50]! CoefficientList[ Series[(1 + z - Log[1/(1 - z)])^(-1), {z, 0, 50}], z]
%o A226226 (PARI) x='x+O('x^66); Vec(serlaplace(1/(1+x-log(1/(1-x))))) \\ _Joerg Arndt_, Jun 01 2013
%Y A226226 The alignments with singletons included are given by A007840.
%K A226226 nonn
%O A226226 0,4
%A A226226 _Ricardo Bittencourt_, May 31 2013