A316989 -id:A316989 - 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

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.

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 24 2018

Let (A,B,d) denote the three-variable bracket polynomial for the two-bridge knot with Conway's notation C(n,k). Then T(n,k) is the leading coefficient of the reduced polynomial x*(1,1,x). In Kauffman's language, T(n,k) is the number of states of the two-bridge knot C(n,k) corresponding to the maximum number of Jordan curves.

			Square array begins:
  1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 1, 1, 1, ...
  1, 1, 3, 2, 2, 2, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  ...

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

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, 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 Maminirina Ramaharo, Illustration of T(2,2)
Franck Maminirina Ramaharo, Note on this sequence and related ones
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial

Cf. A300453, A300454, A316989, A321126, A321127.

b[n_] = If[n == 0 || n == 2, 1, 0];
T[n_, k_] = b[n + k] - (2*b[n]*b[k] + 1)*b[n*k] + b[n] + b[k] + 1;
Table[T[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten

b(n) := if n = 0 or n = 2 then 1 else 0$ /* A154272(n+1) */
T(n, k) := b(n + k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1$
create_list(T(k, n - k), n, 0, 12, k, 0, n);

T(n,0) = T(0,n) = 1, and T(n,k) = b(n+k) - b(n)*b(k) - b(n*k) + c(n)*c(k) for n >= 1, k >= 1, where b(n) = A154272(n+1) and c(n) = A294619(n).
T(n,1) = A300453(n+1,A321126(n,1)).
T(n,2) = A300454(n,A321126(n,2)).
T(n,n) = A321127(n,A004280(n+1)).
G.f.: (1 + (x - x^2)*y - (x - 3*x^2 + x^3)*y^2 - x^2*y^3)/((1 - x)*(1 - y)).
E.g.f.: ((x^2 + 2*exp(x))*exp(y) - x^2 + (2*x - x^2)*y - (1 + x - exp(x))*y^2)/2.

A321126 T(n,k) = max(n + k - 1, n + 1, k + 1), square array read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 4, 5, 6, 7, 6, 5, 5, 5, 6, 7, 8, 7, 6, 6, 6, 6, 7, 8, 9, 8, 7, 7, 7, 7, 7, 8, 9, 10, 9, 8, 8, 8, 8, 8, 8, 9, 10, 11, 10, 9, 9, 9, 9, 9, 9, 9, 10, 11, 12, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13

Offset: 0

Franck Maminirina Ramaharo, Nov 21 2018

T(n,k) - 1 is the maximum degree of d in the three-variable bracket polynomial (A,B,d) for the two-bridge knot with Conway's notation C(n,k). Hence, T(n,k) is the maximum number of Jordan curves that are obtained by splitting the crossings of such knot diagram.

			Square array begins:
    1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
    2,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
    3,  3,  3,  4,  5,  6,  7,  8,  9, 10, ...
    4,  4,  4,  5,  6,  7,  8,  9, 10, 11, ...
    5,  5,  5,  6,  7,  8,  9, 10, 11, 12, ...
    6,  6,  6,  7,  8,  9, 10, 11, 12, 13, ...
    7,  7,  7,  8,  9, 10, 11, 12, 13, 14, ...
    8,  8,  8,  9, 10, 11, 12, 13, 14, 15, ...
    9,  9,  9, 10, 11, 12, 13, 14, 15, 16, ...
   10, 10, 10, 11, 12, 13, 14, 15, 16, 17, ...
  ...

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

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, 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 Maminirina Ramaharo, Illustration of T(2,2)
Franck Maminirina Ramaharo, Note on sequence A321125 and related ones
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial

T(n,1) = degree of the (n+1)-th row polynomial in A300453.
T(n,k) = degree of the n-th row polynomials in A300454 and A321127, k = 2,n, respectively.
Cf. A321125, A316989.
Cf. A003056, A003983, A051125.

Table[Max[k + 1, n - 1, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten

create_list(max(k + 1, n - 1, n - k + 1), n, 0, 10, k, 0, n);

T(n,k) = T(k,n).
T(n,k) = A051125(n+1,k+1) for 0 <= k <= 2, n >= 0, and T(n,k) =  A051125(n+1,k+1) + A003983(n-2,k-2) for k >= 3, n >= 3.
T(n,n) = A004280(n+1).
G.f.: (1 - (2*x - x^2)*y + (x - 2*x^2 + x^3)*y^2 + (x^2 - x^3)*y^3)/(((1 - x)*(1 - y))^2).

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A321125 T(n,k) = b(n+k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A321126 T(n,k) = max(n + k - 1, n + 1, k + 1), square array read by antidiagonals (n >= 0, k >= 0).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

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.

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.

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 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).