A324433 -id:A324433 - 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).

A324450 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals six.

Original entry on oeis.org

1, 28, 832, 21510, 534908, 13859131, 376302024, 10758388849, 324301515064, 10305945419380, 345043568421088, 12157186490730430, 450167342192163864, 17492055607930262356, 712111923015875566112, 30325363998402690968142, 1348761584058475514807527

Offset: 6

Alois P. Heinz, Feb 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..150

Column k=6 of A324429.

Cf. A324432, A324433.

a(n) = A324432(n) - A324433(n).

A324451 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals seven.

Original entry on oeis.org

1, 46, 1973, 68201, 2098472, 64097193, 2008704765, 65104369974, 2195744407327, 77260041066718, 2838770935775962, 108941687605845691, 4365240928102864872, 182511493727454453315, 7955490810231969340127, 361168545333805402618994, 17059236243411156603252558

Offset: 7

Alois P. Heinz, Feb 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..97

Column k=7 of A324429.

Cf. A324433, A324434.

a(n) = A324433(n) - A324434(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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A324450 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals six.

Views

Author

Keywords

Links

Crossrefs

Formula

A324451 Number of labeled cyclic chord diagrams with n chords such that the minimal chord length equals seven.

Views

Author

Keywords

Links

Crossrefs

Formula