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

0,5

Comments

Let <K>(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*<K>(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.

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

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.

Example

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, ...
  ...

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

Prog

(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);

Crossrefs

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

Sequence in context: A326976 A117358 A294333 * A205617 A359888 A204112

Adjacent sequences: A321122 A321123 A321124 * A321126 A321127 A321128

Keyword

nonn,easy,tabl

Author

Franck Maminirina Ramaharo, Nov 24 2018

Status

approved