A300184 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (2*x + 2)^n + (x^2 - 1)*(x + 2)^n.

0, 0, 1, 0, 1, 2, 1, 0, 4, 7, 4, 1, 0, 12, 26, 19, 6, 1, 0, 32, 88, 88, 39, 8, 1, 0, 80, 272, 360, 230, 71, 10, 1, 0, 192, 784, 1312, 1140, 532, 123, 12, 1, 0, 448, 2144, 4368, 4872, 3164, 1162, 211, 14, 1, 0, 1024, 5632, 13568, 18592, 15680, 8176, 2480, 367, 16, 1, 0, 2304, 14336, 39936, 65088, 67872, 46368, 20304, 5262, 655, 18, 1

Offset: 0

Author

Franck Maminirina Ramaharo, Feb 27 2018

Keywords

• nonn
• tabf

Comments

Let L(n;x) = x*(2*x + 2)^n. Then T(n,k) is obtained from the expansion of the polynomial P(n;x) = (x + 2)*P(n-1;x) + L(n-1;x), with P(0;x) = x^2.

Let an n-chain link be the planar diagram that consists of n unknotted circles, linked together in a closed chain. Then T(n,k) is the number of diagrams having k components that are obtained by smoothing each double point (crossing). Kauffman defines the 'smoothing' of a framed 4-graph at a vertex v as "any of the two framed 4-graphs obtained by removing v and repasting the edges" (see links).

Examples

			The triangle T(n,k) begins:
n\k  0     1      2      3      4      5      6      7     8    9  10 11
0:   0     0      1
1:   0     1      2      1
2:   0     4      7      4      1
3:   0    12     26     19      6      1
4:   0    32     88     88     39      8      1
5:   0    80    272    360    230     71     10      1
6:   0   192    784   1312   1140    532    123     12     1
7:   0   448   2144   4368   4872   3164   1162    211    14    1
8:   0  1024   5632  13568  18592  15680   8176   2480   367   16   1
9:   0  2304  14336  39936  65088  67872  46368  20304  5262  655  18  1
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened
• James Kaiser, Jessica S. Purcell, Clint Rollins, Volumes of chain links, arXiv:1107.2865 [math.GT], 2011.
• Louis H. Kauffman, Introductory Lectures on Knot Theory, Selected Lectures Presented at the Advanced School and Conference on Knot Theory and Its Applications to Physics and Biology, World Scientific, 2012.
• Louis H. Kauffman and Vassily O. Manturov, New Ideas in Low Dimensional Topology, World Scientific, 2015.
• Carolina Medina, Jorge Ramírez-Alfonsín and Gelasio Salazar, On the number of unknot diagrams, arXiv:1710.06470 [math.CO], 2017.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
• Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
• Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Crossrefs

Row sums: A000302 (powers of 4).

Cf. A299989, A300192, A300453, A300454, A316659, A316989.

Programs

Mathematica

With[{nmax = 15}, CoefficientList[CoefficientList[Series[(x^2 + y*x/(1 - y*(2*x + 2)))/(1 - y*(x + 2)), {x, 0, nmax + 2}, {y, 0, nmax}], y], x]] // Flatten (* G. C. Greubel, Oct 18 2018 *)
Table[SeriesCoefficient[x^2*(x+2)^n + x*Sum[(x+2)^(n-j-1)*(2*x+2)^j, {j, 0, n-1}], {x, 0, k}], {n, 0, 10}, {k, 0, n+2}]//Flatten  (* Michael De Vlieger, Oct 20 2018 *)

Maxima

T(n, k) := ratcoef((2*x + 2)^n + (x^2 - 1)*(x + 2)^n, x, k)$
create_list(T(n, k), n, 0, 10, k, 0, n + 2);

PARI

{T(n,k) = if(k==0, 0, if(k==1, n*2^(n-1), if(k==n+2, 1, T(n-1, k-1) + 2*T(n-1,k) + 2^(n-1)*binomial(n-1,k-1) )))};
for(n=0,10, for(k=0,n+2, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 20 2018

Formula

T(n,0) = 0, T(n,1) = n*2^(n-1), T(0,2) = 1 and T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + A038208(n-1,k-1).

T(n,1) = A001787(n).

T(n,n) = A295077(n).

T(n,n+1) = A005843(n).

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

Extensions

New name by Franck Maminirina Ramaharo, Oct 17 2018