cp's OEIS Frontend

A259834 Number of permutations of [n] with no fixed points where the maximal displacement of an element equals n-1.

%I A259834 #35 May 30 2019 14:01:51
%S A259834 0,0,1,2,5,20,97,574,3973,31520,281825,2803418,30704101,367114252,
%T A259834 4757800705,66432995030,994204132517,15875195019224,269397248811073,
%U A259834 4841453414347570,91856764780324165,1834779993945449348,38485629141294791201,845788826477292504302
%N A259834 Number of permutations of [n] with no fixed points where the maximal displacement of an element equals n-1.
%C A259834 a(n) counts permutations p of [n] such that p(i) <> i and (p(1) = n or p(n) = 1).
%H A259834 Alois P. Heinz, <a href="/A259834/b259834.txt">Table of n, a(n) for n = 0..450</a>
%F A259834 a(n) = ((2*n^2-11*n+13)*a(n-1) + (2*n-5)*(n-3)*a(n-2))/(2*n-7) for n > 2.
%F A259834 a(n) = (n-2)! * [x^(n-2)] exp(-x)*(x+1)/(x-1)^2 for n > 1.
%F A259834 a(n) ~ 2 * (n-1)! / exp(1). - _Vaclav Kotesovec_, Jul 07 2015
%F A259834 a(n) = y(n,n), n > 1, where y(m+1,n) = (n-m)*y(m,n) + (n-m)*y(m-1,n), with y(0,n)=0, y(1,n)=y(2,n)=1 for all n. - _Benedict W. J. Irwin_, Nov 03 2016
%F A259834 From _Peter Luschny_, Oct 05 2017: (Start)
%F A259834 a(n) = (Gamma(n-1, -1) + 2*Gamma(n, -1)) / e for n >= 2.
%F A259834 a(n) = A000166(n-2) + 2*A000166(n-1) for n >= 2.
%F A259834 a(n) = (2*n-1)*A000166(n-2) - 2*(-1)^n for n >= 2.
%F A259834 a(n) = A000255(n-2) + A000166(n-1) for n >= 2.
%F A259834 a(n+2) = (-1)^n*(-hypergeom([1,1-n], [], 1) + hypergeom([2,2-n], [], 1)) = A292898(2, n) for n >= 0.
%F A259834 a(n) ~ 2*sqrt(2*Pi)*exp(-n-1)*n^(n-1/2). (End)
%F A259834 a(n+2) = A306455(n) + n! for n >= 0. - _Alois P. Heinz_, Feb 16 2019
%e A259834 a(2) = 1: 21.
%e A259834 a(3) = 2: 231, 312.
%e A259834 a(4) = 5: 2341, 3421, 4123, 4312, 4321.
%p A259834 a:= proc(n) option remember; `if`(n<3, [0, 0, 1][n+1],
%p A259834      ((2*n^2-11*n+13)*a(n-1) +(2*n-5)*(n-3)*a(n-2))/(2*n-7))
%p A259834     end:
%p A259834 seq(a(n), n=0..30);
%t A259834 Join[{0, 0}, Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == (n - m)*y[m] + (n - m) y[m - 1], y[0] == 0, y[1] == 1, y[2] == 1}]][n], {n, 2, 30}]] (* _Benedict W. J. Irwin_, Nov 03 2016 *)
%t A259834 Table[If[n<2, 0, Subfactorial[n-2]+2*Subfactorial[n-1]], {n,0,23}] (* _Peter Luschny_, Oct 04 2017 *)
%o A259834 (Python)
%o A259834 def A259834_list(len):
%o A259834     L, u, x, y = [0], 1, 0, 0
%o A259834     for n in range(len):
%o A259834         y, x, u = x, x*n + u, -u
%o A259834         L.append(y + 2*x)
%o A259834     L[1] = 0
%o A259834     return L
%o A259834 print(A259834_list(23)) # _Peter Luschny_, Oct 04 2017
%Y A259834 A diagonal of A259784.
%Y A259834 Cf. A000142, A000166, A000255, A292897, A292898, A306455.
%K A259834 nonn,easy
%O A259834 0,4
%A A259834 _Alois P. Heinz_, Jul 06 2015