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A316659 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n), where v = 3 + 2*x and w = sqrt(5 + 4*x).

%I A316659 #12 May 02 2021 06:27:59
%S A316659 0,0,0,1,0,1,2,1,0,5,8,3,0,16,30,16,2,0,45,104,81,24,2,0,121,340,356,
%T A316659 170,35,2,0,320,1068,1411,932,315,48,2,0,841,3262,5209,4396,2079,532,
%U A316659 63,2,0,2205,9760,18281,18784,11440,4144,840,80,2,0,5776,28746
%N A316659 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n), where v = 3 + 2*x and w = sqrt(5 + 4*x).
%C A316659 The triangle is related to the Kauffman bracket polynomial for the Turk's Head Knot ((3,n)-torus knot). Column 1 matches the determinant of the Turk's Head Knots THK(3,k) A004146.
%H A316659 Louis H. Kauffman, <a href="https://doi.org/10.1090/S0002-9947-1990-0958895-7 ">An invariant of regular isotopy</a>, Trans. Amer. Math. Soc., Vol. 318 (1990), 417-471.
%H A316659 Seong Ju Kim, R. Stees, and L. Taalman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Stees/stees4.html">Sequences of Spiral Knot Determinants</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
%H A316659 Alexander Stoimenow, <a href="http://dx.doi.org/10.4310/CAG.2005.v13.n3.a5">Square numbers, spanning trees and invariants of achiral knots</a>, Communications in Analysis and Geometry, Vol. 13 (2005), 591-631.
%F A316659 T(n,1) = A004146(n).
%F A316659 T(n,2) = A122076(n,1) = A099920(2*n-1).
%F A316659 G.f.: (x^3 - 2*x)/(1 - y) + (2*x - 3*x*y - 2*x^2*y)/(1 - 3*y - 2*x*y + y^2 + 2*x*y^2 + x^2*y^2).
%e A316659 The triangle T(n,k) begins:
%e A316659 n\k: 0      1      2       3       4       5       6      7      8    9  10 11
%e A316659 0:   0      0      0       1
%e A316659 1:   0      1      2       1
%e A316659 2:   0      5      8       3
%e A316659 3:   0     16     30      16       2
%e A316659 4:   0     45    104      81      24       2
%e A316659 5:   0    121    340     356     170      35       2
%e A316659 6:   0    320   1068    1411     932     315      48      2
%e A316659 7:   0    841   3262    5209    4396    2079     532     63      2
%e A316659 8:   0   2205   9760   18281   18784   11440    4144    840     80    2
%e A316659 9:   0   5776  28746   61786   74838   55809   26226   7602   1260   99   2
%e A316659 10:  0  15125  83620  202841  282980  249815  144488  54690  13080 1815 120  2
%e A316659 ...
%t A316659 v = 3 + 2*x; w = Sqrt[5 + 4*x];
%t A316659 row[n_] := CoefficientList[x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n), x];
%t A316659 Array[row, 15, 0] // Flatten
%o A316659 (Maxima)
%o A316659 v : 3 + 2*x$ w : sqrt(5 + 4*x)$
%o A316659 p(n, x) := expand(x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n))$
%o A316659 for n:0 thru 15 do print(makelist(ratcoef(p(n, x), x, k), k, 0, max(3, n + 1)));
%Y A316659 Row sums: A000302 (Powers of 4).
%Y A316659 Row 1: row 1 of A300184, A300192 and row 0 of A300454.
%Y A316659 Row 2: row 2 of A300454.
%Y A316659 Cf. A137396, A299989, A300453.
%K A316659 nonn,tabf
%O A316659 0,7
%A A316659 _Franck Maminirina Ramaharo_, Jul 09 2018