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A324428
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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.
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12
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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T(n,k) = Sum_{j=k..n} A324429(n,j).
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EXAMPLE
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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;
...
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MAPLE
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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);
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MATHEMATICA
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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]];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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