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

0,5

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.

Links

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

Formula

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

Example

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;
  ...

Maple

b:= proc(n, f, m, l, j) option remember; (k-> `if`(n<add(i, i=f)+m+
      add(i, i=l), 0, `if`(n=0, 1, add(`if`(f[i]=0, 0, b(n-1,
      subsop(i=0, f), m+l[1], [subsop(1=[][], l)[], 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],
      [subsop(1=[][], l)[], 0], max(0, j-1)))+b(n-1, f, m+l[1],
      [subsop(1=[][], l)[], 1], max(0, j-1)))))(nops(l))
    end:
A:= (n, k)-> `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 *)

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.

Sequence in context: A185285 A268434 A010107 * A119830 A268435 A039910

Adjacent sequences: A324426 A324427 A324428 * A324430 A324431 A324432

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 27 2019

Status

approved