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

Author

Alois P. Heinz, Feb 27 2019

Keywords

• nonn
• tabl

Comments

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.

Examples

			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;
  ...
		

Links

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

Crossrefs

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.

Programs

Maple

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);

Mathematica

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 *)

Formula

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