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A306386 Number of chord diagrams with n chords all having arc length at least 3.

%I A306386 #36 Feb 27 2019 17:49:16
%S A306386 1,0,0,1,7,68,837,11863,189503,3377341,66564396,1439304777,
%T A306386 33902511983,864514417843,23735220814661,698226455579492,
%U A306386 21914096529153695,731009183350476805,25829581529376423945,963786767538027630275,37871891147795243899204,1563295398737378236910447
%N A306386 Number of chord diagrams with n chords all having arc length at least 3.
%C A306386 A cyclical form of A190823.
%C A306386 Also the number of 2-uniform set partitions of {1...2n} such that, when the vertices are arranged uniformly around a circle, no block has its two vertices separated by an arc length of less than 3.
%H A306386 Alois P. Heinz, <a href="/A306386/b306386.txt">Table of n, a(n) for n = 0..404</a>
%H A306386 Gus Wiseman, <a href="/A306386/a306386_1.png">The a(5) = 68 chord diagrams with all arc lengths at least 3.</a>
%F A306386 a(n) is even <=> n in { A135042 }. - _Alois P. Heinz_, Feb 27 2019
%e A306386 The a(8) = 7 2-uniform set partitions with all arc lengths at least 3:
%e A306386   {{1,4},{2,6},{3,7},{5,8}}
%e A306386   {{1,4},{2,7},{3,6},{5,8}}
%e A306386   {{1,5},{2,6},{3,7},{4,8}}
%e A306386   {{1,5},{2,6},{3,8},{4,7}}
%e A306386   {{1,5},{2,7},{3,6},{4,8}}
%e A306386   {{1,6},{2,5},{3,7},{4,8}}
%e A306386   {{1,6},{2,5},{3,8},{4,7}}
%p A306386 a:= proc(n) option remember; `if`(n<8, [1, 0$2, 1, 7, 68, 837, 11863][n+1],
%p A306386       ((8*n^4-64*n^3+142*n^2-66*n+109)    *a(n-1)
%p A306386       -(24*n^4-248*n^3+870*n^2-1106*n+241)*a(n-2)
%p A306386       +(24*n^4-264*n^3+982*n^2-1270*n+145)*a(n-3)
%p A306386       -(8*n^4-96*n^3+374*n^2-486*n+33)    *a(n-4)
%p A306386       -(4*n^3-24*n^2+39*n-2)              *a(n-5))/(4*n^3-36*n^2+99*n-69))
%p A306386     end:
%p A306386 seq(a(n), n=0..23);  # _Alois P. Heinz_, Feb 27 2019
%t A306386 dtui[{},_]:={{}};dtui[set:{i_,___},n_]:=Join@@Function[s,Prepend[#,s]&/@dtui[Complement[set,s],n]]/@Table[{i,j},{j,Switch[i,1,Select[set,3<#<n-1&],2,Select[set,4<#<n&],_,Select[set,#>i+2&]]}];
%t A306386 Table[Length[dtui[Range[n],n]],{n,0,12,2}]
%Y A306386 Cf. A000296, A000699, A001006, A001147, A001610, A003436, A038041, A054726, A135042, A170941, A190823, A278990, A306419, A322402, A324011, A324169.
%Y A306386 Column k=3 of A324428.
%K A306386 nonn
%O A306386 0,5
%A A306386 _Gus Wiseman_, Feb 26 2019
%E A306386 a(10)-a(16) from _Alois P. Heinz_, Feb 26 2019
%E A306386 a(17)-a(21) from _Alois P. Heinz_, Feb 27 2019