OFFSET
0,7
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.
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.
FORMULA
EXAMPLE
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
...
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
PROG
(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)));
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Franck Maminirina Ramaharo, Jul 09 2018
STATUS
approved