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).

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, 170, 35, 2, 0, 320, 1068, 1411, 932, 315, 48, 2, 0, 841, 3262, 5209, 4396, 2079, 532, 63, 2, 0, 2205, 9760, 18281, 18784, 11440, 4144, 840, 80, 2, 0, 5776, 28746

Offset: 0

Author

Franck Maminirina Ramaharo, Jul 09 2018

Keywords

• nonn
• tabf

Comments

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.

Examples

			The triangle T(n,k) begins:
n\k: 0      1      2       3       4       5       6      7      8    9  10 11
0:   0      0      0       1
1:   0      1      2       1
2:   0      5      8       3
3:   0     16     30      16       2
4:   0     45    104      81      24       2
5:   0    121    340     356     170      35       2
6:   0    320   1068    1411     932     315      48      2
7:   0    841   3262    5209    4396    2079     532     63      2
8:   0   2205   9760   18281   18784   11440    4144    840     80    2
9:   0   5776  28746   61786   74838   55809   26226   7602   1260   99   2
10:  0  15125  83620  202841  282980  249815  144488  54690  13080 1815 120  2
...
		

Links

• Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., Vol. 318 (1990), 417-471.
• Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
• Alexander Stoimenow, Square numbers, spanning trees and invariants of achiral knots, Communications in Analysis and Geometry, Vol. 13 (2005), 591-631.

Crossrefs

Row sums: A000302 (Powers of 4).

Row 1: row 1 of A300184, A300192 and row 0 of A300454.

Row 2: row 2 of A300454.

Cf. A137396, A299989, A300453.

Programs

Mathematica

v = 3 + 2*x; w = Sqrt[5 + 4*x];
row[n_] := CoefficientList[x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n), x];
Array[row, 15, 0] // Flatten

Maxima

v : 3 + 2*x$ w : sqrt(5 + 4*x)$
p(n, x) := expand(x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n))$
for n:0 thru 15 do print(makelist(ratcoef(p(n, x), x, k), k, 0, max(3, n + 1)));

Formula

T(n,1) = A004146(n).

T(n,2) = A122076(n,1) = A099920(2*n-1).

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).