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 A300454 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. 8
 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, 0, 13, 44, 71, 70, 42, 14, 2, 0, 15, 58, 113, 140, 112, 56, 16, 2, 0, 17, 74, 169, 252, 252, 168, 72, 18, 2, 0, 19, 92, 241, 420, 504, 420, 240, 90, 20, 2, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums of column 1,2 and 3 yields {4, 8, 16, 30, 52, ...}, in A046127. Almost twice Pascal's triangle A028326 (up to horizontal shift), except column 0 to 3. 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*(A,B,x), where (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). REFERENCES Inga Johnson and Allison K. Henrich, An Interactive Introduction to Knot Theory, Dover Publications, Inc., 2017. LINKS Agnijo Banerjee, Knot theory. Răzvan Gelca and Fumikazu Nagasato,Some results about the kauffman bracket skein module of the twist knot exterior, J. Knot Theory Ramifications 15 (2006), 1095-1106. L. 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: Iss. 2, Article 7 (2013). Franck Ramaharo, Enumerating the states of the twist knot, arXiv preprint arXiv:1712.06543 [math.CO], 2017. Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018. Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019. Alexander Stoimenow, Generating functions, Fibonacci numbers and rational knots, Journal of Algebra, Volume 310, Issue 2 (2007), 491-525. Eric Weisstein's World of Mathematics,Bracket Polynomial. Wikipedia, Twist knot. FORMULA T(n,1) = A005408(n). T(n,2) = A014206(n). T(n,3) = A064999(n+1). T(n,1) + T(n,2) = A002061(n+2). T(n,1) + T(n,3) = A046127(n+1). T(n,2) + T(n,3) = A155753(n+1). T(n,1) + T(n,2) + T(n,3) = A046127(n+2). T(n,k) = A028326(n,k-1), k >= 4 and n >= k - 1. T(n,k) = A300454(n,k-1) + 2*A300454(n,k) + A007318(n,k-1), with T(n,0) = 0. G.f: (2*x + 2)/(1 - y*(x + 1)) + (x^3 + 2*x^2 - x - 2)/(1 - y). EXAMPLE The triangle T(n,k) begins n\k  0   1    2    3     4     5     6     7     8     9    10   11   12  13 14 0:   0   1    2    1 1:   0   3    4    1 2:   0   5    8    3 3:   0   7   14    9     2 4:   0   9   22   21    10     2 5:   0  11   32   41    30    12     2 6:   0  13   44   71    70    42    14     2 7:   0  15   58  113   140   112    56    16     2 8:   0  17   74  169   252   252   168    72    18     2 9:   0  19   92  241   420   504   420   240    90    20     2 10:  0  21  112  331   660   924   924   660   330   110    22    2 11:  0  23  134  441   990  1584  1848  1584   990   440   132   24    2 12:  0  25  158  573  1430  2574  3432  3432  2574  1430   572  156   26   2 13:  0  27  184  729  2002  4004  6006  6864  6006  4004  2002  728  182  28  2 PROG (Maxima) P(n, x) := 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2\$ T : []\$ for i:0 thru 20 do   T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(3, i + 1)))\$ T; (PARI) row(n) = Vecrev(2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2); tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018 CROSSREFS Row sums: A020707(Pisot sequences). Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300453 ((2,n)-torus knot). Cf. A002061, A005408, A007318, A014206, A028326, A028326, A046127, A046127, A046127, A064999, A155753, A299989, A300454, A300454. Sequence in context: A095884 A128908 A285072 * A155112 A256130 A257566 Adjacent sequences:  A300451 A300452 A300453 * A300455 A300456 A300457 KEYWORD nonn,tabf AUTHOR Franck Maminirina Ramaharo, Mar 06 2018 STATUS approved

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Last modified April 10 05:04 EDT 2020. Contains 333392 sequences. (Running on oeis4.)