A002777 Restricted permutations. (Formerly M3526 N1432)

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

Offset

0,5

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)

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

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 *)

Crossrefs

Cf. A003471.

Sequence in context: A279361 A374982 A003471 * A280923 A118997 A337839

Adjacent sequences: A002774 A002775 A002776 * A002778 A002779 A002780

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved