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

0,6

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).

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.

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)).

Example

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

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 *)

Prog

(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

Crossrefs

Row sums: A000302 (powers of 4).

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

Sequence in context: A266867 A151852 A300864 * A160168 A077929 A178039

Adjacent sequences: A300181 A300182 A300183 * A300185 A300186 A300187

Keyword

nonn,tabf

Author

Franck Maminirina Ramaharo, Feb 27 2018

Extensions

New name by Franck Maminirina Ramaharo, Oct 17 2018

Status

approved