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A075852 Number of permutations s of {1,2,...,n} such that |s(i)-i|>3 for each i=1,2,...,n.

%I A075852 #26 Oct 30 2022 18:19:59
%S A075852 1,0,0,0,0,0,0,0,1,16,436,6984,114124,1799688,29125117,486980182,
%T A075852 8490078104,154750897552,2951968964768,58917663227568,
%U A075852 1229367602071416,26787823838035750,608794318333169289,14411810690642972432
%N A075852 Number of permutations s of {1,2,...,n} such that |s(i)-i|>3 for each i=1,2,...,n.
%C A075852 a(n) equals the permanent of the n X n matrix with 0's along the main diagonal, the subdiagonal, the superdiagonal, the sub-subdiagonal, the super-superdiagonal, the sub-sub-subdiagonal, the super-super-superdiagonal, and 1's everywhere else. - _John M. Campbell_, Jul 09 2011
%H A075852 Alois P. Heinz, <a href="/A075852/b075852.txt">Table of n, a(n) for n = 0..450</a>
%H A075852 George Spahn and Doron Zeilberger, <a href="https://arxiv.org/abs/2108.11285">Automatic Counting of Generalized Latin Rectangles and Trapezoids</a>, arXiv:2108.11285 [math.CO], 2021.
%p A075852 b:= proc(s) option remember; (n-> `if`(n=0, 1, add(
%p A075852       `if`(abs(n-i)>3, b(s minus {i}), 0), i=s)))(nops(s))
%p A075852     end:
%p A075852 a:= n-> b({$1..n}):
%p A075852 seq(a(n), n=0..15);  # _Alois P. Heinz_, Jan 25 2019
%t A075852 a[0] = 1; a[n_] := a[n] = If[n < 8, 0, SparseArray[{Band[{1, 1}] -> 0, Band[{2, 1}] -> 0, Band[{3, 1}] -> 0, Band[{4, 1}] -> 0, Band[{1, 2}] -> 0, Band[{1, 3}] -> 0, Band[{1, 4}] -> 0}, {n, n}, 1] // Permanent];
%t A075852 Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* _Jean-François Alcover_, May 01 2019 *)
%Y A075852 Cf. A001883, A075851.
%K A075852 nonn
%O A075852 0,10
%A A075852 _Reiner Martin_, Oct 15 2002
%E A075852 More terms from _Vladimir Baltic_, _Vladeta Jovovic_, Jan 04 2003
%E A075852 a(21) from _Alois P. Heinz_, Jul 04 2015
%E A075852 a(22)-a(23) from _Alois P. Heinz_, Jan 22 2019
%E A075852 a(0)=1 prepended by _Alois P. Heinz_, Jan 25 2019