A293961 -id:A293961 - cp's OEIS frontend

A293881 Number T(n,k) of linear 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.

1, 0, 1, 0, 2, 1, 0, 10, 4, 1, 0, 69, 26, 9, 1, 0, 616, 230, 79, 19, 1, 0, 6740, 2509, 854, 252, 39, 1, 0, 87291, 32422, 11105, 3441, 796, 79, 1, 0, 1305710, 484180, 167273, 52938, 14296, 2468, 159, 1, 0, 22149226, 8203519, 2855096, 919077, 265103, 59520, 7564, 319, 1

Offset: 0

Alois P. Heinz, Oct 18 2017

Conjecture: column k>0 is asymptotic to (exp(-k+1) - exp(-k)) * 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Oct 25 2017

			Triangle T(n,k) begins:
  1;
  0,       1;
  0,       2,      1;
  0,      10,      4,      1;
  0,      69,     26,      9,     1;
  0,     616,    230,     79,    19,     1;
  0,    6740,   2509,    854,   252,    39,    1;
  0,   87291,  32422,  11105,  3441,   796,   79,   1;
  0, 1305710, 484180, 167273, 52938, 14296, 2468, 159,  1;
  ...

Columns k=0-10 give: A000007, A293914, A293915, A293916, A293917, A293918, A293919, A293920, A293921, A293922, A293923.
Row sums give A001147.
T(2n,n) gives A290688.
Main diagonal and first lower diagonal give: A000012, A054135 (for n>0).
Cf. A293157, A293961.

A293960 Number A(n,k) of linear chord diagrams having n chords and no chord length larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 5, 1, 0, 1, 1, 3, 15, 11, 1, 0, 1, 1, 3, 15, 35, 21, 1, 0, 1, 1, 3, 15, 105, 103, 43, 1, 0, 1, 1, 3, 15, 105, 315, 343, 85, 1, 0, 1, 1, 3, 15, 105, 945, 1141, 979, 171, 1, 0, 1, 1, 3, 15, 105, 945, 3465, 4779, 2915, 341, 1, 0

Offset: 0

Alois P. Heinz, Oct 20 2017

All terms in columns k > 0 are odd.

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,     1,     1,      1, ...
  0, 1,  1,   1,    1,     1,     1,      1, ...
  0, 1,  3,   3,    3,     3,     3,      3, ...
  0, 1,  5,  15,   15,    15,    15,     15, ...
  0, 1, 11,  35,  105,   105,   105,    105, ...
  0, 1, 21, 103,  315,   945,   945,    945, ...
  0, 1, 43, 343, 1141,  3465, 10395,  10395, ...
  0, 1, 85, 979, 4779, 14857, 45045, 135135, ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened

Columns k=0-10 give: A000007, A000012, A001045(n+1), A293995, A293996, A293997, A293998, A293999, A294000, A294001, A294002.
Main diagonal gives A001147.
Cf. A293961.

A(n,k) = Sum_{j=0..k} A293961(n,j).

A(n,k) = A(n,n) = A001147(n) for k >= n.

A293962 Number of linear chord diagrams having n chords and maximal chord length n, a(0)=1.

Original entry on oeis.org

1, 1, 2, 10, 70, 630, 6930, 90090, 1351350, 22972950, 436486050, 9166207050, 210822762150, 5270569053750, 142305364451250, 4126855569086250, 127932522641673750, 4221773247175233750, 147762063651133181250, 5467196355091927706250, 213220657848585180543750

Offset: 0

Alois P. Heinz, Oct 20 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..404
Camille Combe and Samuele Giraudo, Cliff operads: a hierarchy of operads on words, arXiv:2106.14552 [math.CO], 2021.

Cf. A001147, A293961.

a:= proc(n) option remember; `if`(n<3, [1, 1, 2][n+1],
      (2*n-1)*a(n-1))
    end:
seq(a(n), n=0..20);

E.g.f.: (x + 1 + 2/sqrt(1-2*x))/3.
a(n) = (2*n-1)*a(n-1) for n > 2, a(0) = a(1) = 1, a(2) = 2.
a(n) = 2/3 * A001147(n) = 2/3 * (2n-1)!! for n>1.
a(n) = A293961(n,n).

A293963 Number of linear chord diagrams having 2n chords and maximal chord length n, a(0)=1.

Original entry on oeis.org

1, 1, 10, 300, 18186, 1838508, 278873238, 59240617044, 16785089537034, 6116748053227032, 2787180218661133026, 1552687010566633714620, 1038232662085139704575042, 820642288423711927490012112, 756912885595843667978969686938, 805677646550729982087279165847716

Offset: 0

Alois P. Heinz, Oct 20 2017

nonn

Cf. A293961.

a(n) = A293961(2n,n).

cp's OEIS Frontend

A293881 Number T(n,k) of linear 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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A293960 Number A(n,k) of linear chord diagrams having n chords and no chord length larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A293962 Number of linear chord diagrams having n chords and maximal chord length n, a(0)=1.

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A293963 Number of linear chord diagrams having 2n chords and maximal chord length n, a(0)=1.

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