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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.
12
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
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
Row sums give A001147.
Main diagonal gives A000012.
T(n+1,n) gives A001610.
Sequence in context: A185285 A268434 A010107 * A119830 A268435 A039910
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 27 2019
STATUS
approved