A300453 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.

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, 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, 0, 10, 46, 120, 210, 252, 210, 120, 45, 10, 1, 0, 11, 56, 165

Offset: 0

Author

Franck Maminirina Ramaharo, Mar 06 2018

Keywords

• nonn
• tabf

Comments

This is essentially the usual Pascal triangle A007318, horizontally shifted and with the first three columns altered.

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.

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.

\ / \_/ \ / \ /

(1) \/ ==> (2) \/ ==> | |

/\ _ /\ | |

/ \ / \ / \ / \

See example for the case n = 2.

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   0   1
1:   0   1   1
2:   0   2   2
3:   0   3   4    1
4:   0   4   7    4     1
5:   0   5  11   10     5     1
6:   0   6  16   20    15     6     1
7:   0   7  22   35    35    21     7     1
8:   0   8  29   56    70    56    28     8     1
9:   0   9  37   84   126   126    84    36     9     1
10:  0  10  46  120   210   252   210   120    45    10     1
11:  0  11  56  165   330   462   462   330   165    55    11    1
12:  0  12  67  220   495   792   924   792   495   220    66   12   1
13:  0  13  79  286   715  1287  1716  1716  1287   715   286   78  13   1
14:  0  14  92  364  1001  2002  3003  3432  3003  2002  1001  364  91  14  1
...
The states of the (2,2)-torus knot (Hopf Link) are the last four diagrams:
                                    ____  ____
                                   /    \/    \
                                  /     /\     \
                                 |     |  |     |
                                 |     |  |     |
                                  \     \/     /
                                   \____/\____/
                           ___    ____         __________
                         /    \  /    \       /    __    \
                        /     /  \     \     /    /  \    \
                       |      |  |      |   |     |  |     |
                       |      |  |      |   |     |  |     |
                        \      \/      /     \     \/     /
                         \_____/\_____/       \____/\____/
      ____    ____        ____    ____        ____________        __________
     /    \  /    \      /    \  /    \      /     __     \      /    __    \
    /     /  \     \    /     /  \     \    /     /  \     \    /    /  \    \
   |     |    |     |  |     |    |     |  |     |    |     |  |    |    |    |
   |     |    |     |  |     |    |     |  |     |    |     |  |    |    |    |
    \     \  /     /    \     \__/     /    \     \  /     /    \    \__/    /
     \____/  \____/      \____________/      \____/  \____/      \__________/
There are 2 diagrams that consist of two components, and 2 diagrams that consist of one component.
		

References

• Colin Adams, The Knot Book, W. H. Freeman and Company, 1994.
• Louis H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11327 (rows 0 <= n <= 150, flattened).
• Agnijo Banerjee, Knot theory [Foil knot family].
• Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten and Ljiljana Radovic, The Theory of Pseudoknots, arXiv preprint arXiv:1210.6934 [math.GT], 2012.
• Abdullah Kopuzlu, Abdulgani Şahin and Tamer Ugur, On polynomials of K(2,n) torus knots, Applied Mathematical Sciences, Vol. 3 (2009), 2899-2910.
• Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
• Franck Ramaharo, Note on sequences A123192, A137396 and A300453, arXiv:1911.04528 [math.CO], 2019.
• Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.
• Wikipedia, Torus knot.
• Xinfei Li, Xin Liu and Yong-Chang Huang, Tackling tangledness of cosmic strings by knot polynomial topological invariants, arxiv preprint arXiv:1602.08804 [hep-th], 2016.

Crossrefs

Row sums: A000079 (powers of 2).

Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300454 (twist knot).

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).

Cf. A001477, A007318, A007318, A152947.

Programs

Mathematica

f[n_] := CoefficientList[ Expand[(x + 1)^n + x^2 - 1], x]; Array[f, 12, 0] // Flatten (* or *)
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 *)

Maxima

P(n, x) := (x + 1)^n + x^2 - 1$
T : []$
for i:0 thru 20 do
  T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(2, i)))$
T;

PARI

row(n) = Vecrev((x + 1)^n + x^2 - 1);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018

Formula

T(n,1) = A001477(n).

T(n,2) = A152947(n).

T(n,k) = A007318(n,k-1), k >= 1.

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).

G.f.: (x^2 + y*x/(1 - y*(x + 1)))/(1 - y).