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A306386 Number of chord diagrams with n chords all having arc length at least 3. 6
1, 0, 0, 1, 7, 68, 837, 11863, 189503, 3377341, 66564396, 1439304777, 33902511983, 864514417843, 23735220814661, 698226455579492, 21914096529153695, 731009183350476805, 25829581529376423945, 963786767538027630275, 37871891147795243899204, 1563295398737378236910447 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A cyclical form of A190823.

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.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..404

Gus Wiseman, The a(5) = 68 chord diagrams with all arc lengths at least 3.

FORMULA

a(n) is even <=> n in { A135042 }. - Alois P. Heinz, Feb 27 2019

EXAMPLE

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

  {{1,4},{2,6},{3,7},{5,8}}

  {{1,4},{2,7},{3,6},{5,8}}

  {{1,5},{2,6},{3,7},{4,8}}

  {{1,5},{2,6},{3,8},{4,7}}

  {{1,5},{2,7},{3,6},{4,8}}

  {{1,6},{2,5},{3,7},{4,8}}

  {{1,6},{2,5},{3,8},{4,7}}

MAPLE

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

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

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

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

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

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

    end:

seq(a(n), n=0..23);  # Alois P. Heinz, Feb 27 2019

MATHEMATICA

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&]]}];

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

CROSSREFS

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

Column k=3 of A324428.

Sequence in context: A297502 A087567 A328046 * A136629 A197525 A133697

Adjacent sequences:  A306383 A306384 A306385 * A306387 A306388 A306389

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 26 2019

EXTENSIONS

a(10)-a(16) from Alois P. Heinz, Feb 26 2019

a(17)-a(21) from Alois P. Heinz, Feb 27 2019

STATUS

approved

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Last modified January 17 23:37 EST 2020. Contains 330995 sequences. (Running on oeis4.)