A002777 Restricted permutations.

1, 0, 0, 0, 4, 16, 80, 672, 4896, 49920, 460032, 5598720, 62584320, 885381120, 11644323840, 187811205120, 2841958748160, 51481298534400, 881192033648640, 17714783352913920, 338434210452602880, 7477275543168614400

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 468.
• Todd Simpson, Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 2.
• T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]. See Vol. 3 page 468. There may have been some confusion here of this sequence with A003471.
• T. Simpson, Letter to N. J. A. Sloane, Mar. 1992.
• T. Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy)

Crossrefs

Cf. A003471.

Programs

Maple

a:= proc(n) option remember; `if`(n<5, [1, 0$3, 4][n+1],
      (n-1)*a(n-1)+2*`if`(n::even, (n-2)*a(n-3), (n-1)*a(n-2)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 27 2020

Mathematica

nmax = 20; b = ConstantArray[0, nmax+1]; b[[1]] = 1; b[[2]] = 0; b[[3]] = 0; b[[4]] = 0; b[[5]] = 4; Do[b[[n+1]] = (n-1)*b[[n]] + If[EvenQ[n],2*(n-2)*b[[n-2]],2*(n-1)*b[[n-1]]], {n, 5, nmax}]; b (* Vaclav Kotesovec, Mar 07 2014 *)

Formula

a(n) = (n-1)*a(n-1) + 2*(n-d)*a(n-e), where (d, e) = (2, 3) if n even, (1, 2) if n odd.

Recurrence (for n>=7): (3*n^2 - 17*n + 23)*a(n) = (3*n^2 - 17*n + 21)*a(n-1) + (3*n^4 - 23*n^3 + 63*n^2 - 74*n + 34)*a(n-2) - 4*(n-3)*(n-2)*a(n-3) + 2*(n-4)*(n-3)*(3*n^2 - 11*n + 9)*a(n-4). - Vaclav Kotesovec, Mar 07 2014

a(n) ~ c * n!, where c = 5*sinh(sqrt(2))/2^(3/2) - 3*cosh(sqrt(2))/2 = 0.15347184510862040153106983922669125715345689997588202335369... - Vaclav Kotesovec, Mar 07 2014, updated Apr 20 2024

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Sep 24 2001