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A306386 Number of chord diagrams with n chords all having arc length at least 3. 6

%I

%S 1,0,0,1,7,68,837,11863,189503,3377341,66564396,1439304777,

%T 33902511983,864514417843,23735220814661,698226455579492,

%U 21914096529153695,731009183350476805,25829581529376423945,963786767538027630275,37871891147795243899204,1563295398737378236910447

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

%C A cyclical form of A190823.

%C 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 Alois P. Heinz, <a href="/A306386/b306386.txt">Table of n, a(n) for n = 0..404</a>

%H Gus Wiseman, <a href="/A306386/a306386_1.png">The a(5) = 68 chord diagrams with all arc lengths at least 3.</a>

%F a(n) is even <=> n in { A135042 }. - _Alois P. Heinz_, Feb 27 2019

%e The a(8) = 7 2-uniform set partitions with all arc lengths at least 3:

%e {{1,4},{2,6},{3,7},{5,8}}

%e {{1,4},{2,7},{3,6},{5,8}}

%e {{1,5},{2,6},{3,7},{4,8}}

%e {{1,5},{2,6},{3,8},{4,7}}

%e {{1,5},{2,7},{3,6},{4,8}}

%e {{1,6},{2,5},{3,7},{4,8}}

%e {{1,6},{2,5},{3,8},{4,7}}

%p a:= proc(n) option remember; `if`(n<8, [1, 0$2, 1, 7, 68, 837, 11863][n+1],

%p ((8*n^4-64*n^3+142*n^2-66*n+109) *a(n-1)

%p -(24*n^4-248*n^3+870*n^2-1106*n+241)*a(n-2)

%p +(24*n^4-264*n^3+982*n^2-1270*n+145)*a(n-3)

%p -(8*n^4-96*n^3+374*n^2-486*n+33) *a(n-4)

%p -(4*n^3-24*n^2+39*n-2) *a(n-5))/(4*n^3-36*n^2+99*n-69))

%p end:

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Feb 27 2019

%t 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 Table[Length[dtui[Range[n],n]],{n,0,12,2}]

%Y Cf. A000296, A000699, A001006, A001147, A001610, A003436, A038041, A054726, A135042, A170941, A190823, A278990, A306419, A322402, A324011, A324169.

%Y Column k=3 of A324428.

%K nonn

%O 0,5

%A _Gus Wiseman_, Feb 26 2019

%E a(10)-a(16) from _Alois P. Heinz_, Feb 26 2019

%E a(17)-a(21) from _Alois P. Heinz_, Feb 27 2019

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Last modified February 23 07:11 EST 2020. Contains 332159 sequences. (Running on oeis4.)