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%I #29 Apr 23 2022 01:35:24
%S 0,1,5,32,254,2414,26746,338568,4820952,76270032,1327263024,
%T 25196689968,518190651744,11476753967184,272339818023984,
%U 6893370154797312,185387657162396544,5279022594143270784,158674547929990485888,5020389181983702415104,166784921186052433648896
%N The total number of components (cycles) in all alignments.
%C An alignment is a sequence of cycles of an n-permutation, cf. A007840.
%H Seiichi Manyama, <a href="/A215916/b215916.txt">Table of n, a(n) for n = 0..417</a>
%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 180.
%F a(n) = Sum_{k=1...n} s(n,k)*k!*k where s(n,k) is the unsigned Stirling number of the first kind (A132393).
%F E.g.f.: log(1/(1-x))/(1-log(1/(1-x)))^2.
%F a(n) ~ n!*n*exp(n)/(exp(1)-1)^(n+2) . - _Vaclav Kotesovec_, Sep 24 2013
%F E.g.f.: Sum_{k>=0} k * (-log(1-x))^k. - _Seiichi Manyama_, Apr 22 2022
%t nn = 20; a = Log[1/(1 - x)];Range[0, nn]! CoefficientList[
%t D[Series[1/(1 - y a), {x, 0, nn}], y] /. y -> 1, x]
%o (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N, k*(-log(1-x))^k)))) \\ _Seiichi Manyama_, Apr 22 2022
%Y Cf. A007840, A132393.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Aug 27 2012
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