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A002777 Restricted permutations.
(Formerly M3526 N1432)
3
1, 0, 0, 0, 4, 16, 80, 672, 4896, 49920, 460032, 5598720, 62584320, 885381120, 11644323840, 187811205120, 2841958748160, 51481298534400, 881192033648640, 17714783352913920, 338434210452602880, 7477275543168614400 (list; graph; refs; listen; history; text; internal format)
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
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, 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 = 0.15347184510862040153106983922669125715345689997588202335369... - Vaclav Kotesovec, Mar 07 2014
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: A020080 A279361 A003471 * A280923 A118997 A337839
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Sep 24 2001
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)