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 A325791 #12 Nov 01 2020 03:53:49
%S A325791 1,1,1,2,4,20,82,252,1074,7912,39552,152680,776094,5550310,30026848,
%T A325791 108376910
%N A325791 Number of necklace permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.
%C A325791 A necklace permutation is a permutation that is either empty or whose first part is the minimum. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. The only circular subsequence of maximum length is the sequence itself, not any rotation of it. For example, the circular subsequences of (1,3,2) are: (), (1), (2), (3), (1,3), (2,1), (3,2), (1,3,2).
%e A325791 The a(1) = 1 through a(5) = 20 permutations:
%e A325791 (1) (1,2) (1,2,3) (1,2,3,4) (1,2,3,4,5)
%e A325791 (1,3,2) (1,3,2,4) (1,2,3,5,4)
%e A325791 (1,4,2,3) (1,2,4,3,5)
%e A325791 (1,4,3,2) (1,2,4,5,3)
%e A325791 (1,2,5,4,3)
%e A325791 (1,3,2,5,4)
%e A325791 (1,3,4,2,5)
%e A325791 (1,3,4,5,2)
%e A325791 (1,3,5,2,4)
%e A325791 (1,3,5,4,2)
%e A325791 (1,4,2,3,5)
%e A325791 (1,4,2,5,3)
%e A325791 (1,4,3,2,5)
%e A325791 (1,4,5,2,3)
%e A325791 (1,4,5,3,2)
%e A325791 (1,5,2,3,4)
%e A325791 (1,5,2,4,3)
%e A325791 (1,5,3,2,4)
%e A325791 (1,5,3,4,2)
%e A325791 (1,5,4,3,2)
%t A325791 subalt[q_]:=Union[ReplaceList[q,{___,s__,___}:>{s}],DeleteCases[ReplaceList[q,{t___,__,u___}:>{u,t}],{}]];
%t A325791 Table[Length[Select[Permutations[Range[n]],#=={}||First[#]==1&&Range[n*(n+1)/2]==Union[Total/@subalt[#]]&]],{n,0,5}]
%Y A325791 Cf. A002033, A008965, A032153, A103295, A126796, A304793, A325684, A325781, A325786, A325788, A325789, A325790.
%K A325791 nonn,more
%O A325791 0,4
%A A325791 _Gus Wiseman_, May 23 2019
%E A325791 a(11)-a(15) from _Bert Dobbelaere_, Nov 01 2020