OFFSET
0,4
COMMENTS
These are the coefficients of the Kauffman bracket polynomial evaluated at the shadow diagram of the two-bridge knot with Conway's notation C(n,n). Hence, T(n,k) gives the corresponding number of Kauffman states having exactly k circles.
REFERENCES
Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..14282 (rows 0 <= n <= 120, flattened).
Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.
Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.
Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial
FORMULA
T(n,k) = 0 if k = 0, n^2 + 1 if k = 1, and C(2*n, k + 1) - 2*(C(n, k + 1) + C(n, k - 1)) otherwise.
T(n,1) = A002522(n).
T(n,2) = A300401(n,n).
T(n,k) = A094527(n,k-n+1) if n + 1 < k < 2*n and n > 2.
G.f.: x*(1 - (1 + x + x^2)*y + (1 + x)*(2 - x^2)*y^2)/((1 - y)*(1 - y - x*y)*(1 - (1 + x)^2*y)).
E.g.f.: (exp((1 + x)^2*y) - (exp(x) + 2*exp((1 + x)*y))*(1 - x^2))/x.
EXAMPLE
Triangle begins:
n\k | 0 1 2 3 4 5 6 7 8 9 11 12
----+----------------------------------------------------
0 | 0 1
1 | 0 2 2
2 | 0 5 8 3
3 | 0 10 24 21 8 1
4 | 0 17 56 80 64 30 8 1
5 | 0 26 110 220 270 220 122 45 10 1
6 | 0 37 192 495 820 952 804 497 220 66 12 1
...
MATHEMATICA
row[n_] := CoefficientList[Expand[((x + 1)^(2*n) + (x^2 - 1)*(2*(x + 1)^n - 1))/x], x]; Array[row, 12, 0] // Flatten
PROG
(Maxima) T(n, k) := if k = 1 then n^2 + 1 else ((4*k - 2*n)/(k + 1))*binomial(n + 1, k) + binomial(2*n, k + 1)$
create_list(T(n, k), n, 0, 12, k, 0, max(2*n - 1, n + 1));
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Franck Maminirina Ramaharo, Nov 19 2018
STATUS
approved