A324435 -id:A324435 - cp's OEIS frontend

A324428 Number T(n,k) of labeled cyclic chord diagrams with n chords such that every chord has length at least k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 15, 4, 1, 105, 31, 7, 1, 945, 293, 68, 11, 1, 10395, 3326, 837, 159, 18, 1, 135135, 44189, 11863, 2488, 381, 29, 1, 2027025, 673471, 189503, 43169, 7601, 879, 47, 1, 34459425, 11588884, 3377341, 822113, 160784, 23559, 2049, 76, 1, 654729075, 222304897, 66564396, 17066007, 3621067, 607897, 72989, 4788, 123, 1

Offset: 1

Alois P. Heinz, Feb 27 2019

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with 1 <= k <= n. T(n,0) = A001147(n), T(0,k) = 1, T(n,k) = 0 for k > n > 0.

			Triangle T(n,k) begins:
        1;
        3,      1;
       15,      4,      1;
      105,     31,      7,     1;
      945,    293,     68,    11,    1;
    10395,   3326,    837,   159,   18,   1;
   135135,  44189,  11863,  2488,  381,  29,  1;
  2027025, 673471, 189503, 43169, 7601, 879, 47, 1;
  ...

Alois P. Heinz, Rows n = 1..19, flattened

Columns k=1-10 give: A001147, A003436, A306386, A324430, A324431, A324432, A324433, A324434, A324435, A324436.
T(n,n-1) gives A000204.
Cf. A293157, A293881, A324429.

b:= proc(n, f, m, l, j) option remember; (k-> `if`(n `if`(n=0 or k<2, doublefactorial(2*n-1),
              b(2*n-k+1, [1$k-1], 0, [0$k-1], k-1)):
seq(seq(T(n, k), k=1..n), n=1..10);

b[n_, f_List, m_, l_List, j_] := b[n, f, m, l, j] = Function[k, If[n < Total[f] + m + Total[l], 0, If[n == 0, 1, Sum[If[f[[i]] == 0, 0, b[n - 1, ReplacePart[f, i -> 0], m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]],{i, Max[1, j + 1], Min[k, n - 1]}] + If[m == 0, 0, m*b[n - 1, f, m - 1 + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]] + b[n - 1, f, m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 1], Max[0, j - 1]]]]][Length[l]];
T[n_, k_] := If[n == 0 || k < 2, 2^(n-1) Pochhammer[3/2, n-1], b[2n-k+1, Table[1, {k-1}], 0, Table[0, {k-1}], k-1]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

T(n,k) = Sum_{j=k..n} A324429(n,j).

A324452 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals eight.

Original entry on oeis.org

1, 75, 4665, 210738, 8209677, 297511656, 10758827632, 395523803515, 14915877669600, 580537472007709, 23391018103358991, 977123638431619844, 42348895992865182235, 1904609547261379880531, 88871688665105062492072, 4300641510854505723773724

Offset: 8

Alois P. Heinz, Feb 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..50

Column k=8 of A324429.

Cf. A324434, A324435.

a(n) = A324434(n) - A324435(n).

A324453 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals nine.

Original entry on oeis.org

1, 122, 10980, 651962, 31844074, 1382524706, 57759507054, 2408367427106, 101604258942813, 4373090645047650, 193104747980308379, 8776137990134208386, 411230178369151616856, 19887402814123893337150, 993118621992840412774504, 51217036176798656934165346

Offset: 9

Alois P. Heinz, Feb 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..30

Column k=9 of A324429.

Cf. A324435, A324436.

a(n) = A324435(n) - A324436(n).

cp's OEIS Frontend

A324428 Number T(n,k) of labeled cyclic chord diagrams with n chords such that every chord has length at least k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A324452 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals eight.

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A324453 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals nine.

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