A321127 - OEIS

%I #12 Apr 10 2019 21:55:17

%S 0,1,0,2,2,0,5,8,3,0,10,24,21,8,1,0,17,56,80,64,30,8,1,0,26,110,220,

%T 270,220,122,45,10,1,0,37,192,495,820,952,804,497,220,66,12,1,0,50,

%U 308,973,2030,3059,3472,3017,2004,1001,364,91,14,1

%N Irregular triangle read by rows: row n gives the coefficients in the expansion of ((x + 1)^(2*n) + (x^2 - 1)*(2*(x + 1)^n - 1))/x.

%C 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.

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

%H Michael De Vlieger, <a href="/A321127/b321127.txt">Table of n, a(n) for n = 0..14282</a> (rows 0 <= n <= 120, flattened).

%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 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 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/2-bridge_knot">2-bridge knot</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bracket_polynomial">Bracket polynomial</a>

%F 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.

%F T(n,1) = A002522(n).

%F T(n,2) = A300401(n,n).

%F T(n,n) = A001791(n) + A005843(n) - A063524(n).

%F T(n,k) = A094527(n,k-n+1) if  n + 1 < k < 2*n  and n > 2.

%F 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)).

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

%e Triangle begins:

%e n\k | 0   1    2    3    4    5    6    7    8   9  11 12

%e ----+----------------------------------------------------

%e   0 | 0   1

%e   1 | 0   2    2

%e   2 | 0   5    8    3

%e   3 | 0  10   24   21    8    1

%e   4 | 0  17   56   80   64   30    8    1

%e   5 | 0  26  110  220  270  220  122   45   10   1

%e   6 | 0  37  192  495  820  952  804  497  220  66  12  1

%e   ...

%t row[n_] := CoefficientList[Expand[((x + 1)^(2*n) + (x^2 - 1)*(2*(x + 1)^n - 1))/x], x]; Array[row, 12, 0] // Flatten

%o (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)$

%o create_list(T(n, k), n, 0, 12, k, 0, max(2*n - 1, n + 1));

%Y Row sums: A000302.

%Y Row 1 is row 2 in A300453.

%Y Row 2 is also row 2 in A300454 and A316659.

%Y Cf. A299989, A300184, A300192, A316989.

%K nonn,easy,tabf

%O 0,4

%A _Franck Maminirina Ramaharo_, Nov 19 2018