A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2*w)) * (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2*x + 4 + w) - (((v - w)/2)^(n - 1)) * (x^2 + 2*x + 4 - w)), where v = x^2 + 4*x + 4 and w = sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12).

0, 1, 1, 0, 4, 7, 4, 1, 0, 15, 40, 42, 23, 7, 1, 0, 56, 201, 306, 262, 140, 48, 10, 1, 0, 209, 943, 1877, 2189, 1672, 881, 325, 82, 13, 1, 0, 780, 4239, 10412, 15368, 15276, 10841, 5660, 2194, 624, 125, 16, 1, 0, 2911, 18506, 54051, 96501, 118175, 105495

Offset

1,5

Comments

T(n,k) is the number of ways to assign horizontal or vertical barriers at each interior construction dot of the 4 X 2n barrier-free Celtic shadow diagram CK_4^(2n) such that the resulting design consists of exactly k connected components.

The n-th row is the coefficients in the expansion of the Kauffman bracket polynomial for the shadow of the Celtic link CK_4^(2n).

Links

Table of n, a(n) for n=1..55.

Roger Antonsen and Laura Taalman, Categorizing Celtic Knot Designs, in Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture, 2021, pp. 87-94.

Jonathan L. Gross and Thomas W. Tucker, A Celtic Framework for Knots and Links, Discrete & Computational Geometry 46 (2011), 86-99.

Franck Ramaharo, The bracket polynomial of the Celtic link shadow CK_4^(2n), arXiv:2508.10410 [math.GT], 2025. See p. 6.

Formula

T(n,1) = A001353(n).

G.f.: x*y*(x + 1)*(1 - x*y) / (1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2).

Example

The triangle T(n,k) begins:
  n\k 0    1     2     3     4     5       6     7     8     9   10   11  12 13 14
  1:  0    1     1
  2:  0    4     7     4     1
  3:  0   15    40    42    23      7      1
  4:  0   56   201   306   262    140     48    10     1
  5:  0  209   943  1877  2189   1672    881   325    82    13    1
  6:  0  780  4239 10412 15368  15276  10841  5660  2194   624  125   16   1
  7:  0 2911 18506 54051 96501 118175 105495 71107 36885 14817 4579 1064 177 19  1
  ...

Mathematica

With[{nmax = 15}, CoefficientList[CoefficientList[Series[x*y*(x + 1)*(1 - x*y)/(1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2), {x, 0, 2*nmax}, {y, 0, nmax}], y], x]] // Flatten

Prog

(Maxima)
nmax: 15$ v: x^2 + 4*x + 4$ w: sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12)$
p(n, x) := expand((1/(2*w))*(x^2 + x)*((((v + w)/2)^(n - 1))*(x^2 + 2*x + 4 + w) - (((v - w)/2)^(n - 1))*(x^2 + 2*x + 4 - w)))$
create_list(ratcoef(p(n, x), x, k), n, 1, nmax, k, 0, 2*n);

Crossrefs

Row sums: A013730.

Cf. A299989, A300192, A300453, A300454, A316659, A316989.

Sequence in context: A301930 A365940 A365945 * A365943 A201412 A272056

Adjacent sequences: A386871 A386872 A386873 * A386875 A386876 A386877

Keyword

nonn,tabf

Author

Franck Maminirina Ramaharo, Aug 06 2025

Status

approved