This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A322859 #15 Sep 18 2019 05:29:02
%S A322859 1,1,2,4,14,52,256,1396,9064,62420,500000,4250180,40738880,410140060,
%T A322859 4572668112,53214384548,676739353112
%N A322859 The number of permutations p of {1,...,n} such that the numbers 2p(1)-1,...,2p(n)-n are all distinct.
%C A322859 If p=(i,j) is a transposition on letters 1,...,n with 1 <= i < j <= n, then the numbers 2p(1)-1, ..., 2p(n)-n are all distinct if and only if either j >= 2i or j > (i+n)/2. It follows that the number b(n) of such permutations equals A000212(n)=floor(n^2/3).
%F A322859 Conjecture: n! ~ n^(1+o(1))*a(n).
%F A322859 Conjecture: (n-2)a(n-1) <= a(n) <= (n-1)a(n-1).
%F A322859 Conjecture: The polynomial a(1)+a(2)x+...+a(n)x^(n-1) is irreducible for all n. Indeed, it seems that the polynomials are irreducible for any permutation of coefficients except for n=7 where the exceptional permutations are (1,7,3,5,4,6) and (1,3,4,6,2).
%e A322859 For n=4, the a(4)=14 permutations are (), (2,4), (2,3,4), (1,4), (1,4,3,2), (1,4,2,3), (1,4)(2,3), (1,2,4,3), (1,2)(3,4), (1,2,3,4), (1,3), (1,3,2), (1,3)(2,4), (1,3,2,4).
%o A322859 (GAP) Number(Filtered(SymmetricGroup(n),p->Number(Unique(List([1..n],i->2*i^p-i)))=n));
%Y A322859 Cf. A000212.
%Y A322859 A099152 counts the permutations of {1,...,n} such that the numbers p(i)-i (or p(i)+i) are distinct for i=1,...,n.
%K A322859 nonn,hard,more
%O A322859 0,3
%A A322859 _M. Farrokhi D. G._, Dec 29 2018
%E A322859 a(15)-a(16) from _Bert Dobbelaere_, Sep 18 2019