A300192 -id:A300192 - cp's OEIS frontend

A300184 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (2x + 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

Franck Maminirina Ramaharo, Feb 27 2018

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).

			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

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.

Row sums: A000302 (powers of 4).

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

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 *)

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);

{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

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)).

New name by Franck Maminirina Ramaharo, Oct 17 2018

A316989 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (x^2 + 4x + 3)(x + 1)^(2n) + (x^2 - 1)(x^2 + 3*x + 3).

Original entry on oeis.org

0, 1, 3, 3, 1, 0, 7, 14, 9, 2, 0, 13, 37, 43, 26, 8, 1, 0, 19, 72, 129, 141, 98, 42, 10, 1, 0, 25, 119, 291, 463, 504, 378, 192, 63, 12, 1, 0, 31, 178, 553, 1156, 1716, 1848, 1452, 825, 330, 88, 14, 1, 0, 37, 249, 939, 2432, 4576, 6435, 6864, 5577, 3432, 1573

Offset: 0

Franck Maminirina Ramaharo, Jul 18 2018

The triangle is related to the Kauffman bracket polynomial evaluated at the shadow diagram of the two-bridge knot with Conway's notation C(2n,3).

			The triangle T(n,k) begins:
n\k| 0   1    2    3     4     5     9     7     8     9    10   11   12  13 14
-------------------------------------------------------------------------------
0  | 0   1    3    3     1
1  | 0   7   14    9     2
2  | 0  13   37   43    26     8     1
3  | 0  19   72  129   141    98    42    10     1
4  | 0  25  119  291   463   504   378   192    63    12     1
5  | 0  31  178  553  1156  1716  1848  1452   825   330    88   14    1
6  | 0  37  249  939  2432  4576  6435  6864  5577  3432  1573  520  117  16  1
...

Ji-Young Ham and Joongul Lee, An explicit formula for the A-polynomial of the knot with Conway’s notation C(2n,3), Journal of Knot Theory and Its Ramifications 25 (2016), 1-9.
Ryo Hanaki, On scannable properties of the original knot from a knot shadow, Topology and its Applications 194 (2015), 296-305.
Bin Lu and Jianyuan K. Zhong, The Kauffman Polynomials of 2-bridge Knots, arXiv:math/0606114 [math.GT], 2006.

Row sums: 4*A004171.

Cf. A093560, A137396, A299989, A300184, A300192, A300453, A300454, A316659.

T := proc (n, k) if k = 1 then 6*n + 1 else binomial(2*n + 3, k + 1) + (binomial(2*n + 1, k)*(2*k - 2*n) + binomial(4, k)*(2*k - 3))/(k + 1) end if end proc:
for n from 0 to 12 do seq(T(n, k), k = 0 .. max(4, 2*(n + 1))) od;

row[n_] := CoefficientList[(x^2 + 4*x + 3)*(x + 1)^(2*n) + (x^2 - 1)*(x^2 + 3*x + 3), x];
Array[row, 12, 0] // Flatten

T(n, k) := binomial(2*n + 3, k + 1) + (binomial(2*n + 1, k)*(2*k - 2*n) + binomial(4, k)*(2*k - 3))/(k + 1) - kron_delta(1, k)$
for n:0 thru 12 do print(makelist(T(n, k), k, 0, max(4, 2*(n + 1))));

T(n,1) = A016921(n) and T(n,k) = C(2*n+3,k+1) + (C(2*n+1,k)*(2*k - 2*n) + C(4,k)*(2*k - 3))/(k + 1) for k > 1.
T(n,2) = A173247(2*n+1) = A300401(2*n,3).
T(n,3) = 2*A099721(n) + 3.
T(n,4) = A244730(n) - A002412(n) + 1.
T(n,k) = A093560(2*n,k) for n > 2 and k > 4.
G.f.: (x^2 + 4*x + 3)/(1 - y*(x + 1)^2) + (x^4 + 3*x^3 + 2*x^2 - 3*x - 3)/(1 - y).

A316659 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial x(x^2 - 2) + x(((v - w)/2)^n + ((v + w)/2)^n), where v = 3 + 2x and w = sqrt(5 + 4x).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 0, 5, 8, 3, 0, 16, 30, 16, 2, 0, 45, 104, 81, 24, 2, 0, 121, 340, 356, 170, 35, 2, 0, 320, 1068, 1411, 932, 315, 48, 2, 0, 841, 3262, 5209, 4396, 2079, 532, 63, 2, 0, 2205, 9760, 18281, 18784, 11440, 4144, 840, 80, 2, 0, 5776, 28746

Offset: 0

Franck Maminirina Ramaharo, Jul 09 2018

The triangle is related to the Kauffman bracket polynomial for the Turk's Head Knot ((3,n)-torus knot). Column 1 matches the determinant of the Turk's Head Knots THK(3,k) A004146.

			The triangle T(n,k) begins:
n\k: 0      1      2       3       4       5       6      7      8    9  10 11
0:   0      0      0       1
1:   0      1      2       1
2:   0      5      8       3
3:   0     16     30      16       2
4:   0     45    104      81      24       2
5:   0    121    340     356     170      35       2
6:   0    320   1068    1411     932     315      48      2
7:   0    841   3262    5209    4396    2079     532     63      2
8:   0   2205   9760   18281   18784   11440    4144    840     80    2
9:   0   5776  28746   61786   74838   55809   26226   7602   1260   99   2
10:  0  15125  83620  202841  282980  249815  144488  54690  13080 1815 120  2
...

Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., Vol. 318 (1990), 417-471.
Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Alexander Stoimenow, Square numbers, spanning trees and invariants of achiral knots, Communications in Analysis and Geometry, Vol. 13 (2005), 591-631.

Row sums: A000302 (Powers of 4).
Row 1: row 1 of A300184, A300192 and row 0 of A300454.
Row 2: row 2 of A300454.
Cf. A137396, A299989, A300453.

v = 3 + 2*x; w = Sqrt[5 + 4*x];
row[n_] := CoefficientList[x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n), x];
Array[row, 15, 0] // Flatten

v : 3 + 2*x$ w : sqrt(5 + 4*x)$
p(n, x) := expand(x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n))$
for n:0 thru 15 do print(makelist(ratcoef(p(n, x), x, k), k, 0, max(3, n + 1)));

T(n,1) = A004146(n).
T(n,2) = A122076(n,1) = A099920(2*n-1).
G.f.: (x^3 - 2*x)/(1 - y) + (2*x - 3*x*y - 2*x^2*y)/(1 - 3*y - 2*x*y + y^2 + 2*x*y^2 + x^2*y^2).

A321127 Irregular triangle read by rows: row n gives the coefficients in the expansion of ((x + 1)^(2n) + (x^2 - 1)(2*(x + 1)^n - 1))/x.

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 5, 8, 3, 0, 10, 24, 21, 8, 1, 0, 17, 56, 80, 64, 30, 8, 1, 0, 26, 110, 220, 270, 220, 122, 45, 10, 1, 0, 37, 192, 495, 820, 952, 804, 497, 220, 66, 12, 1, 0, 50, 308, 973, 2030, 3059, 3472, 3017, 2004, 1001, 364, 91, 14, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 19 2018

These are the coefficients of the Kauffman bracket polynomial evaluated at the shadow diagram of the two-bridge knot with Conway's notation C(n,n). Hence, T(n,k) gives the corresponding number of Kauffman states having exactly k circles.

			Triangle begins:
n\k | 0   1    2    3    4    5    6    7    8   9  11 12
----+----------------------------------------------------
  0 | 0   1
  1 | 0   2    2
  2 | 0   5    8    3
  3 | 0  10   24   21    8    1
  4 | 0  17   56   80   64   30    8    1
  5 | 0  26  110  220  270  220  122   45   10   1
  6 | 0  37  192  495  820  952  804  497  220  66  12  1
  ...

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Michael De Vlieger, Table of n, a(n) for n = 0..14282 (rows 0 <= n <= 120, flattened).
Louis 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 (2013), 98-113.
Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial

Row sums: A000302.
Row 1 is row 2 in A300453.
Row 2 is also row 2 in A300454 and A316659.
Cf. A299989, A300184, A300192, A316989.

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

T(n, k) := if k = 1 then n^2 + 1 else  ((4*k - 2*n)/(k + 1))*binomial(n + 1, k) + binomial(2*n, k + 1)$
create_list(T(n, k), n, 0, 12, k, 0, max(2*n - 1, n + 1));

T(n,k) = 0 if k = 0, n^2 + 1 if k = 1, and C(2*n, k + 1) - 2*(C(n, k + 1) + C(n, k - 1)) otherwise.
T(n,1) = A002522(n).
T(n,2) = A300401(n,n).
T(n,n) = A001791(n) + A005843(n) - A063524(n).
T(n,k) = A094527(n,k-n+1) if  n + 1 < k < 2*n  and n > 2.
G.f.: x*(1 - (1 + x + x^2)*y + (1 + x)*(2 - x^2)*y^2)/((1 - y)*(1 - y - x*y)*(1 - (1 + x)^2*y)).
E.g.f.: (exp((1 + x)^2*y) - (exp(x) + 2*exp((1 + x)*y))*(1 - x^2))/x.

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2w)) (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2x + 4 + w) - (((v - w)/2)^(n - 1)) (x^2 + 2x + 4 - w)), where v = x^2 + 4x + 4 and w = sqrt(x^4 + 4x^3 + 12x^2 + 20*x + 12).

Original entry on oeis.org

0, 1, 1, 0, 4, 7, 4, 1, 0, 15, 40, 42, 23, 7, 1, 0, 56, 201, 306, 262, 140, 48, 10, 1, 0, 209, 943, 1877, 2189, 1672, 881, 325, 82, 13, 1, 0, 780, 4239, 10412, 15368, 15276, 10841, 5660, 2194, 624, 125, 16, 1, 0, 2911, 18506, 54051, 96501, 118175, 105495

Offset: 1

Franck Maminirina Ramaharo, Aug 06 2025

T(n,k) is the number of ways to assign horizontal or vertical barriers at each interior construction dot of the 4 X 2n barrier-free Celtic shadow diagram CK_4^(2n) such that the resulting design consists of exactly k connected components.

The n-th row is the coefficients in the expansion of the Kauffman bracket polynomial for the shadow of the Celtic link CK_4^(2n).

			The triangle T(n,k) begins:
  n\k 0    1     2     3     4     5       6     7     8     9   10   11  12 13 14
  1:  0    1     1
  2:  0    4     7     4     1
  3:  0   15    40    42    23      7      1
  4:  0   56   201   306   262    140     48    10     1
  5:  0  209   943  1877  2189   1672    881   325    82    13    1
  6:  0  780  4239 10412 15368  15276  10841  5660  2194   624  125   16   1
  7:  0 2911 18506 54051 96501 118175 105495 71107 36885 14817 4579 1064 177 19  1
  ...

Roger Antonsen and Laura Taalman, Categorizing Celtic Knot Designs, in Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture, 2021, pp. 87-94.
Jonathan L. Gross and Thomas W. Tucker, A Celtic Framework for Knots and Links, Discrete & Computational Geometry 46 (2011), 86-99.
Franck Ramaharo, The bracket polynomial of the Celtic link shadow CK_4^(2n), arXiv:2508.10410 [math.GT], 2025. See p. 6.

Row sums: A013730.

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

With[{nmax = 15}, CoefficientList[CoefficientList[Series[x*y*(x + 1)*(1 - x*y)/(1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2), {x, 0, 2*nmax}, {y, 0, nmax}], y], x]] // Flatten

nmax: 15$ v: x^2 + 4*x + 4$ w: sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12)$
p(n, x) := expand((1/(2*w))*(x^2 + x)*((((v + w)/2)^(n - 1))*(x^2 + 2*x + 4 + w) - (((v - w)/2)^(n - 1))*(x^2 + 2*x + 4 - w)))$
create_list(ratcoef(p(n, x), x, k), n, 1, nmax, k, 0, 2*n);

T(n,1) = A001353(n).

G.f.: x*y*(x + 1)*(1 - x*y) / (1 - ((x + 2)^2)*y + ((x + 1)^3)*y^2).

cp's OEIS Frontend

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.

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A316989 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (x^2 + 4*x + 3)*(x + 1)^(2*n) + (x^2 - 1)*(x^2 + 3*x + 3).

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A316659 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial x*(x^2 - 2) + x*(((v - w)/2)^n + ((v + w)/2)^n), where v = 3 + 2*x and w = sqrt(5 + 4*x).

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A321127 Irregular triangle read by rows: row n gives the coefficients in the expansion of ((x + 1)^(2*n) + (x^2 - 1)*(2*(x + 1)^n - 1))/x.

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A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

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A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2*w)) * (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2*x + 4 + w) - (((v - w)/2)^(n - 1)) * (x^2 + 2*x + 4 - w)), where v = x^2 + 4*x + 4 and w = sqrt(x^4 + 4*x^3 + 12*x^2 + 20*x + 12).

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

A316989 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (x^2 + 4x + 3)(x + 1)^(2n) + (x^2 - 1)(x^2 + 3*x + 3).

A316659 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial x(x^2 - 2) + x(((v - w)/2)^n + ((v + w)/2)^n), where v = 3 + 2x and w = sqrt(5 + 4x).

A321127 Irregular triangle read by rows: row n gives the coefficients in the expansion of ((x + 1)^(2n) + (x^2 - 1)(2*(x + 1)^n - 1))/x.

A386874 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (1/(2w)) (x^2 + x) * ((((v + w)/2)^(n - 1)) * (x^2 + 2x + 4 + w) - (((v - w)/2)^(n - 1)) (x^2 + 2x + 4 - w)), where v = x^2 + 4x + 4 and w = sqrt(x^4 + 4x^3 + 12x^2 + 20*x + 12).