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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).

%I A386874 #12 Aug 19 2025 10:57:57
%S A386874 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,
%T A386874 943,1877,2189,1672,881,325,82,13,1,0,780,4239,10412,15368,15276,
%U A386874 10841,5660,2194,624,125,16,1,0,2911,18506,54051,96501,118175,105495
%N 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).
%C A386874 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.
%C A386874 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).
%H A386874 Roger Antonsen and Laura Taalman, <a href="https://archive.bridgesmathart.org/2021/bridges2021-87.html#gsc.tab=0">Categorizing Celtic Knot Designs</a>, in Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture, 2021, pp. 87-94.
%H A386874 Jonathan L. Gross and Thomas W. Tucker, <a href="https://doi.org/10.1007/s00454-010-9257-0">A Celtic Framework for Knots and Links</a>, Discrete & Computational Geometry 46 (2011), 86-99.
%H A386874 Franck Ramaharo, <a href="https://arxiv.org/abs/2508.10410">The bracket polynomial of the Celtic link shadow CK_4^(2n)</a>, arXiv:2508.10410 [math.GT], 2025. See p. 6.
%F A386874 T(n,1) = A001353(n).
%F A386874 G.f.: x*y*(x + 1)*(1 - x*y) / (1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2).
%e A386874 The triangle T(n,k) begins:
%e A386874   n\k 0    1     2     3     4     5       6     7     8     9   10   11  12 13 14
%e A386874   1:  0    1     1
%e A386874   2:  0    4     7     4     1
%e A386874   3:  0   15    40    42    23      7      1
%e A386874   4:  0   56   201   306   262    140     48    10     1
%e A386874   5:  0  209   943  1877  2189   1672    881   325    82    13    1
%e A386874   6:  0  780  4239 10412 15368  15276  10841  5660  2194   624  125   16   1
%e A386874   7:  0 2911 18506 54051 96501 118175 105495 71107 36885 14817 4579 1064 177 19  1
%e A386874   ...
%t A386874 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
%o A386874 (Maxima)
%o A386874 nmax: 15$ v: x^2 + 4*x + 4$ w: sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12)$
%o A386874 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)))$
%o A386874 create_list(ratcoef(p(n, x), x, k), n, 1, nmax, k, 0, 2*n);
%Y A386874 Row sums: A013730.
%Y A386874 Cf. A299989, A300192, A300453, A300454, A316659, A316989.
%K A386874 nonn,tabf
%O A386874 1,5
%A A386874 _Franck Maminirina Ramaharo_, Aug 06 2025