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
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
KEYWORD
nonn,tabf
AUTHOR
Franck Maminirina Ramaharo, Feb 27 2018
EXTENSIONS
New name by Franck Maminirina Ramaharo, Oct 17 2018
STATUS
approved