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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 (list; table; graph; refs; listen; history; text; internal format)
 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 `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

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Last modified September 19 09:52 EDT 2024. Contains 376008 sequences. (Running on oeis4.)