A002777 - OEIS

%I M3526 N1432 #38 Apr 20 2024 15:18:26

%S 1,0,0,0,4,16,80,672,4896,49920,460032,5598720,62584320,885381120,

%T 11644323840,187811205120,2841958748160,51481298534400,

%U 881192033648640,17714783352913920,338434210452602880,7477275543168614400

%N Restricted permutations.

%D T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 468.

%D Todd Simpson, Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A002777/b002777.txt">Table of n, a(n) for n = 0..200</a>

%H T. Muir, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM9350.0003.001">The Theory of Determinants in the Historical Order of Development</a>, 4 vols., Macmillan, NY, 1906-1923, Vol. 2.

%H T. Muir, <a href="/A003471/a003471.pdf">The Theory of Determinants in the Historical Order of Development</a>, 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.

%H T. Simpson, <a href="/A002777/a002777.pdf">Letter to N. J. A. Sloane, Mar. 1992</a>.

%H T. Simpson, <a href="/A007016/a007016.pdf">Permutations with unique fixed and reflected points</a>, Preprint. (Annotated scanned copy)

%F 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.

%F 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

%F 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

%p a:= proc(n) option remember; `if`(n<5, [1, 0$3, 4][n+1],

%p       (n-1)*a(n-1)+2*`if`(n::even, (n-2)*a(n-3), (n-1)*a(n-2)))

%p     end:

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Jun 27 2020

%t 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 *)

%Y Cf. A003471.

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_

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