A324431 -id:A324431 - 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).

A324448 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals four.

Original entry on oeis.org

1, 10, 141, 2107, 35568, 661329, 13444940, 297333278, 7122103435, 183969320652, 5102519835609, 151340386825771, 4782068352314304, 160422533608810186, 5695524314835911655, 213389274860898491690, 8414796730663230017112, 348422910620718837979244

Offset: 4

Alois P. Heinz, Feb 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..404

Column k=4 of A324429.

Cf. A324430, A324431.

a(n) = A324430(n) - A324431(n).

A324449 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals five.

Original entry on oeis.org

1, 17, 352, 6722, 137225, 3013170, 70909863, 1785795023, 48031299838, 1376860759082, 41968192206641, 1356883954535010, 46416330714749286, 1675869323349039899, 63713356727612022712, 2544933663663025077386, 106578973284507947559875, 4670547639650409700162167

Offset: 5

Alois P. Heinz, Feb 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..200

Column k=5 of A324429.

Cf. A324431, A324432.

a(n) = A324431(n) - A324432(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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A324448 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals four.

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A324449 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals five.

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