A293881 -id:A293881 - cp's OEIS frontend

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.

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

Offset: 1

Alois P. Heinz, Feb 27 2019

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.

			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;
  ...

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

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.

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

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]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

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

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.

Original entry on oeis.org

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

Alois P. Heinz, Feb 27 2019

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.

			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;
  ...

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

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.

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);

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 *)

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

A293157 Triangle read by rows: T(n,k) = number of linear chord diagrams with n chords such that every chord has length at least k (1 <= k <= n).

Original entry on oeis.org

1, 3, 1, 15, 5, 1, 105, 36, 10, 1, 945, 329, 99, 20, 1, 10395, 3655, 1146, 292, 40, 1, 135135, 47844, 15422, 4317, 876, 80, 1, 2027025, 721315, 237135, 69862, 16924, 2628, 160, 1, 34459425, 12310199, 4106680, 1251584, 332507, 67404, 7884, 320, 1, 654729075, 234615096, 79154927, 24728326, 6944594, 1627252, 269616, 23652, 640, 1

Offset: 1

N. J. A. Sloane, Oct 10 2017

There is a surprising change in notation in Sullivan (2016) between Definition 1 and Table 1.

The first 11 columns are given in the reference.

			Triangle begins:
      1;
      3,    1;
     15,    5,    1;
    105,   36,   10,    1;
    945,  329,   99,   20,    1;
  10395, 3655, 1146,  292,   40,    1;
  ...

Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.

Cf. A000806, A001147, A190823, A190824, A190825, A020714, A278990, A293156, A293881.

More terms from Alois P. Heinz, Oct 17 2017

A293961 Number T(n,k) of linear chord diagrams having n chords and maximal chord length k (or k=0 if n=0); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 10, 0, 1, 10, 24, 70, 0, 1, 20, 82, 212, 630, 0, 1, 42, 300, 798, 2324, 6930, 0, 1, 84, 894, 3800, 10078, 30188, 90090, 0, 1, 170, 2744, 18186, 51804, 150046, 452724, 1351350, 0, 1, 340, 8594, 71624, 313006, 851692, 2545390, 7695828, 22972950

Offset: 0

Alois P. Heinz, Oct 20 2017

All terms in columns k > 1 are even.

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   2;
  0, 1,   4,   10;
  0, 1,  10,   24,    70;
  0, 1,  20,   82,   212,   630;
  0, 1,  42,  300,   798,  2324,   6930;
  0, 1,  84,  894,  3800, 10078,  30188,  90090;
  0, 1, 170, 2744, 18186, 51804, 150046, 452724, 1351350;
  ...

Alois P. Heinz, Rows n = 0..20, flattened

Columns k=0-2 give: A000007, A057427, A167030(n+1).
Row sums give A001147.
Main diagonal gives A293962.
T(2n,n) gives A293963.
Cf. A293881, A293960.

A(n,k) = A293960(n,k) - A293960(n,k-1) for k>0, A(n,0) = A000007(n).

A293914 Number of linear chord diagrams having n chords and minimal chord length one.

Original entry on oeis.org

1, 2, 10, 69, 616, 6740, 87291, 1305710, 22149226, 420113979, 8810083360, 202397302184, 5054993327385, 136370982998354, 3951907646699026, 122432049983991285, 4038019273764782056, 141261207847788852620, 5224370049771189478371, 203670050053447478737214

Offset: 1

Alois P. Heinz, Oct 19 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..404

Column k=1 of A293881.

a:= proc(n) option remember; `if`(n<3, n, (4*n-6)*
      a(n-1)-2*(2*n^2-8*n+7)*a(n-2)-(2*n-5)*a(n-3))
    end:
seq(a(n), n=1..25);

a(n) ~ (1 - exp(-1)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 25 2017

A293915 Number of linear chord diagrams having n chords and minimal chord length two.

Original entry on oeis.org

1, 4, 26, 230, 2509, 32422, 484180, 8203519, 155460169, 3257843351, 74802301553, 1867393802229, 50358879172771, 1458899632505052, 45185432509804438, 1489952528266230695, 52112346134820625126, 1926974225717684659004, 75110765705496454871866

Offset: 2

Alois P. Heinz, Oct 19 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..404

Column k=2 of A293881.

Recurrence: (24*n^2 - 182*n + 339)*a(n) = (96*n^3 - 800*n^2 + 1894*n - 991)*a(n-1) - (96*n^4 - 824*n^3 + 1804*n^2 + 650*n - 3571)*a(n-2) + 2*(144*n^4 - 1788*n^3 + 8032*n^2 - 15489*n + 10821)*a(n-3) - 2*(144*n^4 - 2028*n^3 + 10452*n^2 - 23337*n + 18994)*a(n-4) + (96*n^4 - 1592*n^3 + 9452*n^2 - 23794*n + 21419)*a(n-5) + (96*n^3 - 1040*n^2 + 3550*n - 3841)*a(n-6) + (24*n^2 - 134*n + 181)*a(n-7). - Vaclav Kotesovec, Oct 25 2017

a(n) ~ (exp(-1) - exp(-2)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 25 2017

A293916 Number of linear chord diagrams having n chords and minimal chord length three.

Original entry on oeis.org

1, 9, 79, 854, 11105, 167273, 2855096, 54426601, 1146050151, 26418261876, 661704149855, 17894827830455, 519672858520861, 16129429179859894, 532839970205194270, 18667063990244959414, 691265187531408182336, 26979801565685043088417, 1106929654165005908000185

Offset: 3

Alois P. Heinz, Oct 19 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..404

Column k=3 of A293881.

a(n) ~ (exp(-2) - exp(-3)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 25 2017

A293917 Number of linear chord diagrams having n chords and minimal chord length four.

Original entry on oeis.org

1, 19, 252, 3441, 52938, 919077, 17783732, 379205917, 8832657552, 223147872970, 6078406509776, 177602480890788, 5541376133638552, 183896037042429100, 6468180930063929404, 240368899912444102612, 9410883311417206598912, 387192534875365889313239

Offset: 4

Alois P. Heinz, Oct 19 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..404

Column k=4 of A293881.

a(n) ~ (exp(-3) - exp(-4)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 25 2017

A293918 Number of linear chord diagrams having n chords and minimal chord length five.

Original entry on oeis.org

1, 39, 796, 14296, 265103, 5317342, 116235247, 2760729582, 70870525294, 1956309944351, 57803029075220, 1820704829013410, 60915564036123607, 2157741151756229543, 80681075613289079236, 3176017007139110899224, 131302895736890434892224, 5688257327967598478393309

Offset: 5

Alois P. Heinz, Oct 19 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..404

Column k=5 of A293881.

a(n) ~ (exp(-4) - exp(-5)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 25 2017

A293919 Number of linear chord diagrams having n chords and minimal chord length six.

Original entry on oeis.org

1, 79, 2468, 59520, 1357636, 31817497, 788783482, 20863949689, 589285383470, 17736904863864, 567344416149454, 19230190154966409, 688795573160438196, 26004353625119063230, 1032308832642494976585, 42994854622890093855000, 1874878774341679655658359

Offset: 6

Alois P. Heinz, Oct 19 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..405

Column k=6 of A293881.

a(n) ~ (exp(-5) - exp(-6)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 25 2017

cp's OEIS Frontend

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.

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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.

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A293157 Triangle read by rows: T(n,k) = number of linear chord diagrams with n chords such that every chord has length at least k (1 <= k <= n).

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A293961 Number T(n,k) of linear chord diagrams having n chords and maximal chord length k (or k=0 if n=0); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A293914 Number of linear chord diagrams having n chords and minimal chord length one.

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A293915 Number of linear chord diagrams having n chords and minimal chord length two.

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A293916 Number of linear chord diagrams having n chords and minimal chord length three.

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A293917 Number of linear chord diagrams having n chords and minimal chord length four.

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A293918 Number of linear chord diagrams having n chords and minimal chord length five.

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