This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A300453 #66 May 12 2020 20:59:06
%S A300453 0,0,1,0,1,1,0,2,2,0,3,4,1,0,4,7,4,1,0,5,11,10,5,1,0,6,16,20,15,6,1,0,
%T A300453 7,22,35,35,21,7,1,0,8,29,56,70,56,28,8,1,0,9,37,84,126,126,84,36,9,1,
%U A300453 0,10,46,120,210,252,210,120,45,10,1,0,11,56,165
%N A300453 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.
%C A300453 This is essentially the usual Pascal triangle A007318, horizontally shifted and with the first three columns altered.
%C A300453 Let P(n;x) = (x + 1)^n + x^2 - 1. Then P(n;x) = P(n-1;x) + x*(x + 1)^(n - 1), with P(0;x) = x^2.
%C A300453 Let a (2,n)-torus knot be projected on the plane. The resulting projection is a planar diagram with n double points. Then, T(n,k) gives the number of state diagrams having k components that are obtained by deleting each double point, then pasting the edges in one of the two ways as shown below.
%C A300453 \ / \___/ \ / \ /
%C A300453 (1) \/ ==> (2) \/ ==> | |
%C A300453 /\ ___ /\ | |
%C A300453 / \ / \ / \ / \
%C A300453 See example for the case n = 2.
%D A300453 Colin Adams, The Knot Book, W. H. Freeman and Company, 1994.
%D A300453 Louis H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
%H A300453 Michael De Vlieger, <a href="/A300453/b300453.txt">Table of n, a(n) for n = 0..11327</a> (rows 0 <= n <= 150, flattened).
%H A300453 Agnijo Banerjee, <a href="http://agnijomaths.com/categories/geometry/topology/knot_theory.html">Knot theory</a> [Foil knot family].
%H A300453 Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten and Ljiljana Radovic, <a href="https://arxiv.org/abs/1210.6934">The Theory of Pseudoknots</a>, arXiv preprint arXiv:1210.6934 [math.GT], 2012.
%H A300453 Abdullah Kopuzlu, Abdulgani Şahin and Tamer Ugur, <a href="https://www.researchgate.net/publication/265777759_On_polynomials_of_K2n_torus_knots">On polynomials of K(2,n) torus knots</a>, Applied Mathematical Sciences, Vol. 3 (2009), 2899-2910.
%H A300453 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 A300453 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 A300453 Franck Ramaharo, <a href="https://arxiv.org/abs/2002.06672">A bracket polynomial for 2-tangle shadows</a>, arXiv:2002.06672 [math.CO], 2020.
%H A300453 Wikipedia, <a href="https://en.wikipedia.org/wiki/Torus_knot">Torus knot</a>.
%H A300453 Xinfei Li, Xin Liu and Yong-Chang Huang, <a href="https://arxiv.org/abs/1602.08804">Tackling tangledness of cosmic strings by knot polynomial topological invariants</a>, arxiv preprint arXiv:1602.08804 [hep-th], 2016.
%F A300453 T(n,1) = A001477(n).
%F A300453 T(n,2) = A152947(n).
%F A300453 T(n,k) = A007318(n,k-1), k >= 1.
%F A300453 T(n,0) = 0, T(0,1) = 1, T(0,2) = 1 and T(n,k) = T(n-1,k) + A007318(n-1,k-1).
%F A300453 G.f.: (x^2 + y*x/(1 - y*(x + 1)))/(1 - y).
%e A300453 The triangle T(n,k) begins
%e A300453 n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e A300453 0: 0 0 1
%e A300453 1: 0 1 1
%e A300453 2: 0 2 2
%e A300453 3: 0 3 4 1
%e A300453 4: 0 4 7 4 1
%e A300453 5: 0 5 11 10 5 1
%e A300453 6: 0 6 16 20 15 6 1
%e A300453 7: 0 7 22 35 35 21 7 1
%e A300453 8: 0 8 29 56 70 56 28 8 1
%e A300453 9: 0 9 37 84 126 126 84 36 9 1
%e A300453 10: 0 10 46 120 210 252 210 120 45 10 1
%e A300453 11: 0 11 56 165 330 462 462 330 165 55 11 1
%e A300453 12: 0 12 67 220 495 792 924 792 495 220 66 12 1
%e A300453 13: 0 13 79 286 715 1287 1716 1716 1287 715 286 78 13 1
%e A300453 14: 0 14 92 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
%e A300453 ...
%e A300453 The states of the (2,2)-torus knot (Hopf Link) are the last four diagrams:
%e A300453 ____ ____
%e A300453 / \/ \
%e A300453 / /\ \
%e A300453 | | | |
%e A300453 | | | |
%e A300453 \ \/ /
%e A300453 \____/\____/
%e A300453 ___ ____ __________
%e A300453 / \ / \ / __ \
%e A300453 / / \ \ / / \ \
%e A300453 | | | | | | | |
%e A300453 | | | | | | | |
%e A300453 \ \/ / \ \/ /
%e A300453 \_____/\_____/ \____/\____/
%e A300453 ____ ____ ____ ____ ____________ __________
%e A300453 / \ / \ / \ / \ / __ \ / __ \
%e A300453 / / \ \ / / \ \ / / \ \ / / \ \
%e A300453 | | | | | | | | | | | | | | | |
%e A300453 | | | | | | | | | | | | | | | |
%e A300453 \ \ / / \ \__/ / \ \ / / \ \__/ /
%e A300453 \____/ \____/ \____________/ \____/ \____/ \__________/
%e A300453 There are 2 diagrams that consist of two components, and 2 diagrams that consist of one component.
%t A300453 f[n_] := CoefficientList[ Expand[(x + 1)^n + x^2 - 1], x]; Array[f, 12, 0] // Flatten (* or *)
%t A300453 CoefficientList[ CoefficientList[ Series[(x^2 + y*x/(1 - y*(x + 1)))/(1 - y), {y, 0, 11}, {x, 0, 11}], y], x] // Flatten (* _Robert G. Wilson v_, Mar 08 2018 *)
%o A300453 (Maxima)
%o A300453 P(n, x) := (x + 1)^n + x^2 - 1$
%o A300453 T : []$
%o A300453 for i:0 thru 20 do
%o A300453 T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(2, i)))$
%o A300453 T;
%o A300453 (PARI) row(n) = Vecrev((x + 1)^n + x^2 - 1);
%o A300453 tabl(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Mar 12 2018
%Y A300453 Row sums: A000079 (powers of 2).
%Y A300453 Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300454 (twist knot).
%Y A300453 When n = 3 (trefoil), the corresponding 4-tuplet (0,3,4,1) appears in several triangles: A030528 (row 5), A256130 (row 3), A128908 (row 3), A186084 (row 6), A284938 (row 7), A101603 (row 3), A155112 (row 3), A257566 (row 3).
%Y A300453 Cf. A001477, A007318, A007318, A152947.
%K A300453 nonn,tabf
%O A300453 0,8
%A A300453 _Franck Maminirina Ramaharo_, Mar 06 2018