A215916 The total number of components (cycles) in all alignments.

0, 1, 5, 32, 254, 2414, 26746, 338568, 4820952, 76270032, 1327263024, 25196689968, 518190651744, 11476753967184, 272339818023984, 6893370154797312, 185387657162396544, 5279022594143270784, 158674547929990485888, 5020389181983702415104, 166784921186052433648896

Offset: 0

Author

Geoffrey Critzer, Aug 27 2012

Keywords

• nonn

Comments

An alignment is a sequence of cycles of an n-permutation, cf. A007840.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..417
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 180.

Crossrefs

Cf. A007840, A132393.

Programs

Mathematica

nn = 20; a = Log[1/(1 - x)];Range[0, nn]! CoefficientList[
  D[Series[1/(1 - y a), {x, 0, nn}], y] /. y -> 1, x]

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

Formula

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).

E.g.f.: log(1/(1-x))/(1-log(1/(1-x)))^2.

a(n) ~ n!*n*exp(n)/(exp(1)-1)^(n+2) . - Vaclav Kotesovec, Sep 24 2013

E.g.f.: Sum_{k>=0} k * (-log(1-x))^k. - Seiichi Manyama, Apr 22 2022