A293156 -id:A293156 - cp's OEIS frontend

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).

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

A321311 Number of linear chord diagrams having n+2 chords and minimal chord length n.

Original entry on oeis.org

10, 26, 79, 252, 796, 2468, 7564, 23012, 69676, 210308, 633484, 1905572, 5726956, 17201348, 51645004, 155016932, 465214636, 1395971588, 4188570124, 12567021092, 37703684716, 113116297028, 339359376844, 1018099102052, 3054339249196, 9163101633668, 27489472673164

Offset: 1

Seiichi Manyama, Nov 17 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5, -6)

A diagonal of A293881.

Cf. A257113, A293156.

Join[{10, 26, 79}, LinearRecurrence[{5, -6}, {252, 796}, 24]] (* Jean-François Alcover, Nov 24 2018 *)

Vec((10 - 24*x + 9*x^2 + 13*x^3 + 10*x^4)/((1 - 2*x)*(1 - 3*x)) + O(x^40)) \\ Andrew Howroyd, Nov 17 2018

a(n) = A293881(n+2,n).
a(n) = 5*a(n-1) - 6*a(n-2) for n > 5.
a(n) = A293156(n) - 5*2^(n-1).
G.f.: x*(10 - 24*x + 9*x^2 + 13*x^3 + 10*x^4)/((1 - 2*x)*(1 - 3*x)). - Andrew Howroyd, Nov 17 2018
2*3^4*a(n) = 2^3*73*3^n-5*3^4*2^n for n>3. - R. J. Mathar, Jan 25 2023

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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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A321311 Number of linear chord diagrams having n+2 chords and minimal chord length n.

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