A078480 - OEIS

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A078480

Number of permutations p of {1,2,...,n} such that |p(i)-i| != 1 for all i.

1, 1, 1, 2, 5, 21, 117, 792, 6205, 55005, 543597, 5922930, 70518905, 910711193, 12678337945, 189252400480, 3015217932073, 51067619064873, 916176426422089, 17355904144773970, 346195850534379613, 7252654441500887309

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OFFSET

0,4

COMMENTS

For positive n, a(n) equals the permanent of the n X n matrix with 0's along the superdiagonal and the subdiagonal, and 1's everywhere else. [John M. Campbell, Jul 09 2011]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 223.

N. S. Mendelsohn, The asymptotic series for a certain class of permutation problems, Canadian Journal of Mathematics, vol. VIII, No.2, 1956, p.238 (Example 5).

FORMULA

G.f.: 1/(1-x^2)*Sum_{n>=0} n!*(x/(1+x)^2)^n. - Vladeta Jovovic, Jun 26 2007

Asymptotic (N. S. Mendelsohn, 1956): a(n)/n! -> 1/e^2

Recurrence: a(n) = n*a(n-1) - (n-2)*a(n-3) - a(n-4), for n>=5

MATHEMATICA

(* Explicit formula: *) Table[Sum[Sum[(-1)^k*(i-k)!*Binomial[2i-k, k], {k, 0, i}], {i, 0, n}], {n, 0, 21}] (* Vaclav Kotesovec, Mar 28 2011 *)

CROSSREFS

Cf. A000179, A000271.