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%I #48 Feb 23 2023 09:21:58
%S 0,1,2,1,0,3,4,1,0,5,8,3,0,7,14,9,2,0,9,22,21,10,2,0,11,32,41,30,12,2,
%T 0,13,44,71,70,42,14,2,0,15,58,113,140,112,56,16,2,0,17,74,169,252,
%U 252,168,72,18,2,0,19,92,241,420,504,420,240,90,20,2,0
%N Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2.
%C Row sums of column 1,2 and 3 yields {4, 8, 16, 30, 52, ...}, in A046127.
%C Almost twice Pascal's triangle A028326 (up to horizontal shift), except column 0 to 3.
%C The polynomial P(n;x) = 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2 is a simplified version of the bracket polynomial associated with a twist knot of n half twists that is only concerned with the enumeration of the state diagrams. The simplification arises when the twist knot is thought of as a planar diagram with no crossing information at each double point. In this case, P(n;x) = x*<T>(A,B,x), where <T>(A,B,d) denotes the bracket polynomial for the n-twist knot (see links for the definition of the bracket polynomial). For example, the bracket polynomial for the trefoil (n = 2) is A^3*d^1 + 3*BA^2*d^0 + 3*AB^2*d^1 + B^3*d^2, where A and B are the "splitting variables". Then setting A = B = 1 and d = x, we obtain 3 + 4*x + x^2 (also see A299989, row 1).
%D Inga Johnson and Allison K. Henrich, An Interactive Introduction to Knot Theory, Dover Publications, Inc., 2017.
%H Agnijo Banerjee, <a href="http://agnijomaths.com/categories/geometry/topology/knot_theory.html">Knot theory</a>.
%H Răzvan Gelca and Fumikazu Nagasato,<a href="https://doi.org/10.1142/S0218216506004968">Some results about the kauffman bracket skein module of the twist knot exterior</a>, J. Knot Theory Ramifications 15 (2006), 1095-1106.
%H L. 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: Iss. 2, Article 7 (2013).
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1712.06543">Enumerating the states of the twist knot</a>, arXiv preprint arXiv:1712.06543 [math.CO], 2017.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1807.05256">A one-variable bracket polynomial for some Turk's head knots</a>, arXiv:1807.05256 [math.CO], 2018.
%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 Alexander Stoimenow, <a href="https://doi.org/10.1016/j.jalgebra.2006.11.031">Generating functions, Fibonacci numbers and rational knots</a>, Journal of Algebra, Volume 310, Issue 2 (2007), 491-525.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BracketPolynomial.html">Bracket Polynomial</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Twist_knot">Twist knot</a>.
%F T(n,1) = A005408(n).
%F T(n,2) = A014206(n).
%F T(n,3) = A064999(n+1).
%F T(n,1) + T(n,2) = A002061(n+2).
%F T(n,1) + T(n,3) = A046127(n+1).
%F T(n,2) + T(n,3) = A155753(n+1).
%F T(n,1) + T(n,2) + T(n,3) = A046127(n+2).
%F T(n,k) = A028326(n,k-1), k >= 4 and n >= k - 1.
%F T(n,k) = A300454(n,k-1) + 2*A300454(n,k) + A007318(n,k-1), with T(n,0) = 0.
%F G.f: (2*x + 2)/(1 - y*(x + 1)) + (x^3 + 2*x^2 - x - 2)/(1 - y).
%e The triangle T(n,k) begins
%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e 0: 0 1 2 1
%e 1: 0 3 4 1
%e 2: 0 5 8 3
%e 3: 0 7 14 9 2
%e 4: 0 9 22 21 10 2
%e 5: 0 11 32 41 30 12 2
%e 6: 0 13 44 71 70 42 14 2
%e 7: 0 15 58 113 140 112 56 16 2
%e 8: 0 17 74 169 252 252 168 72 18 2
%e 9: 0 19 92 241 420 504 420 240 90 20 2
%e 10: 0 21 112 331 660 924 924 660 330 110 22 2
%e 11: 0 23 134 441 990 1584 1848 1584 990 440 132 24 2
%e 12: 0 25 158 573 1430 2574 3432 3432 2574 1430 572 156 26 2
%e 13: 0 27 184 729 2002 4004 6006 6864 6006 4004 2002 728 182 28 2
%o (Maxima)
%o P(n, x) := 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2$
%o T : []$
%o for i:0 thru 20 do
%o T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(3, i + 1)))$
%o T;
%o (PARI) row(n) = Vecrev(2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2);
%o tabl(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Mar 12 2018
%Y Row sums: A020707(Pisot sequences).
%Y Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300453 ((2,n)-torus knot).
%Y Cf. A002061, A005408, A007318, A014206, A028326, A028326, A046127, A046127, A046127, A064999, A155753, A299989, A300454, A300454.
%K nonn,tabf
%O 0,3
%A _Franck Maminirina Ramaharo_, Mar 06 2018
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