OFFSET
0,4
COMMENTS
An n-swap move consists of the removal of n edges and addition of n different edges which result in a new tour. The type can be characterized by how the n segments of the original tour formed by the removal are reassembled.
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..100
Helsgaun, Keld, General k-opt submoves for the Lin-Kernighan TSP heuristic, Math. Program. Comput. 1, No. 2-3, 119-163 (2009).
FORMULA
a(n) = (-1)^n + Sum_{i=0..n-1} (-1)^(n-1-i)*binomial(n,i+1)*i!*2^i = (-1)^n + A120765(n).
E.g.f.: exp(-x)*(1-log(1-2*x)/2)
a(n) ~ (n-1)! * 2^(n-1) * exp(-1/2). - Vaclav Kotesovec, Oct 08 2013
MATHEMATICA
m = 18; CoefficientList[ Series[ Exp[-x]*(1 - Log[1-2x]/2), {x, 0, m}], x]*Range[0, m]! (* Jean-François Alcover, Jul 25 2011, after g.f. *)
PROG
(PARI) { for (n=0, 100, a=(-1)^n + sum(i=0, n-1, (-1)^(n-1-i)*binomial(n, i+1)*i!*2^i); write("b061714.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 26 2009
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
David Applegate, Jun 21 2001
EXTENSIONS
Revised by Max Alekseyev, Jul 03 2006
STATUS
approved