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 A123192 Triangle read by rows: row n gives the coefficients in the expansion of x^abs(3*n - 2)*p(n;x), where p(n;x) denotes the bracket polynomial for the (2,n)-torus knots. 3

%I

%S -1,0,0,0,-1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,-1,1,0,0,0,-1,0,0,0,0,0,0,0,

%T -1,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,-1,0,0,0,1,0,0,0,-1,

%U 0,0,0,0,0,0,0,-1,-1,0,0,0,1,0,0,0,-1,0,0,0

%N Triangle read by rows: row n gives the coefficients in the expansion of x^abs(3*n - 2)*p(n;x), where p(n;x) denotes the bracket polynomial for the (2,n)-torus knots.

%C From _Franck Maminirina Ramaharo_, Aug 11 2018: (Start)

%C Using Kauffman's notation, the formal expression of the bracket polynomial for the (2,n)-torus knot is defined as follows:

%C K(n;A,B,d) = A*K(n-1;A,B,d) + B*(A + B*d)^(n - 1) with K(0;A,B,d) = d.

%C - The polynomial in this sequence is defined as p(n;x) = K(n;x,1/x,-x^2-x^(-2)), and verifies p(n;x) = x*p(n-1;x) + (-1)^(n - 1)*x^(-3*n + 2).

%C - The polynomial x*K(n;1,1,x) yields (x + 1)^n + x^2 - 1 which is the bracket evaluated at the shadow diagram of the (2,n)-torus knot, see A300453.

%C - The polynomial sqrt(x)*K(n;-1,sqrt(x),sqrt(x)) yields (x - 1)^n + (x - 1)*(-1)^n. This is the chromatic polynomial for the n-cycle graph which is the medial graph of the (2,n)-torus knot, see A137396.

%C The planar diagram of the (2,0)-torus knot is two non-interesecting circles.

%C (End)

%D Louis H. Kauffman, Knots and Physics (Third Edition), World Scientific, 2001. See p. 38 and p. 353.

%H Paul Corbitt, <a href="http://educ.jmu.edu/~taalmala/OJUPKT/">Torus Links and the Bracket Polynomial</a>.

%H 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 Franck Ramaharo, <a href="https://arxiv.org/abs/1911.04528">Note on sequences A123192, A137396 and A300453</a>, arXiv:1911.04528 [math.CO], 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BracketPolynomial.html">Bracket Polynomial</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusKnot.html">Torus Knot</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Torus_knot">Torus knot</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Knot_(mathematics)#Medial_graph">Medial graph</a>.

%e From _Franck Maminirina Ramaharo_, Aug 11 2018: (Start)

%e The bracket polynomial for some value of n:

%e p(0;x) = -x^2 - 1/x^2;

%e p(1;x) = -x^3;

%e p(2;x) = -x^4 - 1/x^4;

%e p(3;x) = -x^5 - 1/x^3 + 1/x^7;

%e p(4;x) = -x^6 - 1/x^2 + 1/x^6 - 1/x^10;

%e p(5;x) = -x^7 - 1/x + 1/x^5 - 1/x^9 + 1/x^13;

%e p(6;x) = -x^8 - 1 + 1/x^4 - 1/x^8 + 1/x^12 - 1/x^16;

%e p(7;x) = -x^9 - x + 1/x^3 - 1/x^7 + 1/x^11 - 1/x^15 + 1/x^19;

%e ...

%e The triangle giving the coefficients in x^abs(3*n - 2)*p(n;x) begins:

%e -1, 0, 0, 0, -1

%e 0, 0, 0, 0, -1

%e -1, 0, 0, 0, 0, 0, 0, 0, -1

%e 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1

%e -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1

%e 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1

%e ...

%e (End)

%o (Maxima)

%o K(n, A, B, d) := if n = 0 then d else A*K(n - 1, A, B, d) + B*(A + B*d)^(n - 1)\$

%o p(n, x) := x^abs(3*n - 2)*K(n, x, 1/x, -x^(-2) - x^2)\$

%o t(n, k) := ratcoef(p(n, x), x, k)\$

%o T:[]\$

%o for n:0 thru 10 do T:append(T, makelist(t(n,k), k, 0, max(4, 4*n)))\$

%o T; /* _Franck Maminirina Ramaharo_, Aug 11 2018 */

%Y Cf. A029694, A051764, A137396, A300453.

%K tabf,sign

%O 0,1

%A _Roger L. Bagula_, Oct 03 2006

%E Partially edited by _N. J. A. Sloane_, May 22 2007

%E Edited, new name, and corrected by _Franck Maminirina Ramaharo_, Aug 11 2018

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Last modified August 1 05:51 EDT 2021. Contains 346384 sequences. (Running on oeis4.)