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*<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).
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.
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).
KEYWORD
nonn,tabf
AUTHOR
Franck Maminirina Ramaharo, Mar 06 2018
STATUS
approved