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.

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

Offset: 0

Author

Franck Maminirina Ramaharo, Mar 06 2018

Keywords

• nonn
• tabf

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).

Examples

			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
		

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.

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.

Programs

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

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).