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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).

%I A321125 #22 Feb 16 2025 08:33:57
%S A321125 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,
%T A321125 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,
%U A321125 2,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1
%N 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).
%C A321125 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.
%D A321125 Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.
%H A321125 Michael De Vlieger, <a href="/A321125/b321125.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened).
%H A321125 Louis H. Kauffman, <a href="https://doi.org/10.1016/0040-9383(87)90009-7">State models and the Jones polynomial</a>, Topology Vol. 26 (1987), 395-407.
%H A321125 Kelsey Lafferty, <a href="https://scholar.rose-hulman.edu/rhumj/vol14/iss2/7/">The three-variable bracket polynomial for reduced, alternating links</a>, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.
%H A321125 Matthew Overduin, <a href="https://www.math.csusb.edu/reu/OverduinPaper.pdf">The three-variable bracket polynomial for two-bridge knots</a>, California State University REU, 2013.
%H A321125 Franck Maminirina Ramaharo, <a href="/A321125/a321125.pdf">Illustration of T(2,2)</a>
%H A321125 Franck Maminirina Ramaharo, <a href="/A321125/a321125.txt">Note on this sequence and related ones</a>
%H A321125 Franck Ramaharo, <a href="https://arxiv.org/abs/1902.08989">A generating polynomial for the two-bridge knot with Conway's notation C(n,r)</a>, arXiv:1902.08989 [math.CO], 2019.
%H A321125 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BracketPolynomial.html">Bracket Polynomial</a>
%H A321125 Wikipedia, <a href="https://en.wikipedia.org/wiki/2-bridge_knot">2-bridge knot</a>
%H A321125 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bracket_polynomial">Bracket polynomial</a>
%F A321125 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).
%F A321125 T(n,1) = A300453(n+1,A321126(n,1)).
%F A321125 T(n,2) = A300454(n,A321126(n,2)).
%F A321125 T(n,n) = A321127(n,A004280(n+1)).
%F A321125 G.f.: (1 + (x - x^2)*y - (x - 3*x^2 + x^3)*y^2 - x^2*y^3)/((1 - x)*(1 - y)).
%F A321125 E.g.f.: ((x^2 + 2*exp(x))*exp(y) - x^2 + (2*x - x^2)*y - (1 + x - exp(x))*y^2)/2.
%e A321125 Square array begins:
%e A321125   1, 1, 1, 1, 1, 1, ...
%e A321125   1, 2, 1, 1, 1, 1, ...
%e A321125   1, 1, 3, 2, 2, 2, ...
%e A321125   1, 1, 2, 1, 1, 1, ...
%e A321125   1, 1, 2, 1, 1, 1, ...
%e A321125   1, 1, 2, 1, 1, 1, ...
%e A321125   ...
%t A321125 b[n_] = If[n == 0 || n == 2, 1, 0];
%t A321125 T[n_, k_] = b[n + k] - (2*b[n]*b[k] + 1)*b[n*k] + b[n] + b[k] + 1;
%t A321125 Table[T[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten
%o A321125 (Maxima) b(n) := if n = 0 or n = 2 then 1 else 0$ /* A154272(n+1) */
%o A321125 T(n, k) := b(n + k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1$
%o A321125 create_list(T(k, n - k), n, 0, 12, k, 0, n);
%Y A321125 Cf. A300453, A300454, A316989, A321126, A321127.
%K A321125 nonn,easy,tabl
%O A321125 0,5
%A A321125 _Franck Maminirina Ramaharo_, Nov 24 2018