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A061714
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Number of types of (n-1)-swap moves for traveling salesman problem. Number of circular permutations on elements 0,1,...,2n-1 where every two elements 2i,2i+1 and no two elements 2i-1,2i are adjacent.
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4
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1, 0, 1, 4, 25, 208, 2121, 25828, 365457, 5895104, 106794993, 2147006948, 47436635753, 1142570789072, 29797622256377, 836527783016196, 25153234375160993, 806519154686509056, 27470342073410272609
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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)
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MATHEMATICA
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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. *)
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PROG
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(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
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CROSSREFS
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Cf. A001171 (sequential n-swap moves).
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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