A321125 T(n,k) = b(n+k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Author

Franck Maminirina Ramaharo, Nov 24 2018

Keywords

• nonn
• easy
• tabl

Comments

Let (A,B,d) denote the three-variable bracket polynomial for the two-bridge knot with Conway's notation C(n,k). Then T(n,k) is the leading coefficient of the reduced polynomial x*(1,1,x). In Kauffman's language, T(n,k) is the number of states of the two-bridge knot C(n,k) corresponding to the maximum number of Jordan curves.

Examples

			Square array begins:
  1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 1, 1, 1, ...
  1, 1, 3, 2, 2, 2, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  ...
		

References

• Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, 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 Maminirina Ramaharo, Illustration of T(2,2)
• Franck Maminirina Ramaharo, Note on this sequence and related ones
• Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
• Eric Weisstein's World of Mathematics, Bracket Polynomial
• Wikipedia, 2-bridge knot
• Wikipedia, Bracket polynomial

Crossrefs

Cf. A300453, A300454, A316989, A321126, A321127.

Programs

Mathematica

b[n_] = If[n == 0 || n == 2, 1, 0];
T[n_, k_] = b[n + k] - (2*b[n]*b[k] + 1)*b[n*k] + b[n] + b[k] + 1;
Table[T[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten

Maxima

b(n) := if n = 0 or n = 2 then 1 else 0$ /* A154272(n+1) */
T(n, k) := b(n + k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1$
create_list(T(k, n - k), n, 0, 12, k, 0, n);

Formula

T(n,0) = T(0,n) = 1, and T(n,k) = b(n+k) - b(n)*b(k) - b(n*k) + c(n)*c(k) for n >= 1, k >= 1, where b(n) = A154272(n+1) and c(n) = A294619(n).

T(n,1) = A300453(n+1,A321126(n,1)).

T(n,2) = A300454(n,A321126(n,2)).

T(n,n) = A321127(n,A004280(n+1)).

G.f.: (1 + (x - x^2)*y - (x - 3*x^2 + x^3)*y^2 - x^2*y^3)/((1 - x)*(1 - y)).

E.g.f.: ((x^2 + 2*exp(x))*exp(y) - x^2 + (2*x - x^2)*y - (1 + x - exp(x))*y^2)/2.