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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).
3
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
0,7
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
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).
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
Row sums: A000302 (Powers of 4).
Row 1: row 1 of A300184, A300192 and row 0 of A300454.
Row 2: row 2 of A300454.
Sequence in context: A140589 A331955 A185209 * A241218 A266904 A299198
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved