A324429 Number T(n,k) of labeled cyclic chord diagrams having n chords and minimal chord length k (or k=0 if n=0); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 11, 3, 1, 0, 74, 24, 6, 1, 0, 652, 225, 57, 10, 1, 0, 7069, 2489, 678, 141, 17, 1, 0, 90946, 32326, 9375, 2107, 352, 28, 1, 0, 1353554, 483968, 146334, 35568, 6722, 832, 46, 1, 0, 22870541, 8211543, 2555228, 661329, 137225, 21510, 1973, 75, 1

Offset: 0

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 0 <= k <= n. T(n,k) = 0 for k > n.

Examples

			Triangle T(n,k) begins:
  1;
  0,       1;
  0,       2,      1;
  0,      11,      3,      1;
  0,      74,     24,      6,     1;
  0,     652,    225,     57,    10,    1;
  0,    7069,   2489,    678,   141,   17,   1;
  0,   90946,  32326,   9375,  2107,  352,  28,  1;
  0, 1353554, 483968, 146334, 35568, 6722, 832, 46, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..19, flattened

Crossrefs

Columns k=0-10 give: A000007, A324445, A324446, A324447, A324448, A324449, A324450, A324451, A324452, A324453, A324454.

Row sums give A001147.

Main diagonal gives A000012.

T(n+1,n) gives A001610.

Cf. A293157, A293881, A324428.

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)):
T:= (n, k)-> `if`(n=k, 1, A(n, k)-A(n, k+1)):
seq(seq(T(n, k), k=0..n), n=0..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]];
A[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]];
T[n_, k_] := If[n == k, 1, A[n, k] - A[n, k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

Formula

T(n,k) = A324428(n,k) - A324428(n,k+1) for k > 0, T(n,0) = A000007(n).