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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. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

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

FORMULA

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

EXAMPLE

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;

  ...

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:

T:= (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)):

seq(seq(T(n, k), k=1..n), n=1..10);

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.

Sequence in context: A072479 A264772 A263917 * A131440 A269950 A190088

Adjacent sequences:  A324425 A324426 A324427 * A324429 A324430 A324431

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Feb 27 2019

STATUS

approved

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Last modified February 20 17:51 EST 2020. Contains 332082 sequences. (Running on oeis4.)