A300453 -id:A300453 - cp's OEIS frontend

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

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).

A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 12, 6, 1, 0, 0, 5, 32, 27, 8, 1, 0, 0, 6, 80, 108, 48, 10, 1, 0, 0, 7, 192, 405, 256, 75, 12, 1, 0, 0, 8, 448, 1458, 1280, 500, 108, 14, 1, 0, 0, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 0, 0, 10, 2304

Offset: 0

Franck Maminirina Ramaharo, Oct 14 2018

T(n,k) is the number of length n*k binary words of n consecutive blocks of length k, respectively, one of the blocks having exactly k letters 1, and the other having exactly one letter 0. First column follows from the next definition.
In Kauffman's language, T(n,k) is the total number of Jordan trails that are obtained by placing state markers at the crossings of the Pretzel universe P(k, k, ..., k) having n tangles, of k half-twists respectively. In other words, T(n,k) is the number of ways of splitting the crossings of the Pretzel knot shadow P(k, k, ..., k) such that the final diagram is a single Jordan curve. The aforementionned binary words encode these operations by assigning each tangle a length k binary words with the adequate choice for splitting the crossings.
Columns are linear recurrence sequences with signature (2*k, -k^2).

			Square array begins:
    0, 0,   0,    0,     0,      0,      0,      0, ...
    1, 1,   1,    1,     1,      1,      1,      1, ...
    0, 2,   4,    6,     8,     10,     12,     14, ... A005843
    0, 3,  12,   27,    48,     75,    108,    147, ... A033428
    0, 4,  32,  108,   256,    500,    864,   1372, ... A033430
    0, 5,  80,  405,  1280,   3125,   6480,  12005, ... A269792
    0, 6, 192, 1458,  6144,  18750,  46656, 100842, ...
    0, 7, 448, 5103, 28672, 109375, 326592, 823543, ...
    ...
T(3,2) = 3*2^(3 - 1) = 12. The corresponding binary words are 110101, 110110, 111001, 111010, 011101, 011110, 101101, 101110, 010111, 011011, 100111, 101011.

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

Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Alexander Stoimenow, Everywhere Equivalent 2-Component Links, Symmetry Vol. 7 (2015), 365-375.
Wikipedia, Pretzel link

Antidiagonal sums: A101495.
Column 1 is column 2 of A300453.
Column 2 is column 1 of A300184.
Cf. A104002, A320530.

T[n_, k_] = If [k > 0, n*k^(n - 1), If[k == 0 && n == 1, 1, 0]];
Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 12}]//Flatten

T(n, k) := if k > 0 then n*k^(n - 1) else if k = 0 and n = 1 then 1 else 0$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, nn))$

T(n,k) = (2*k)*T(n-1,k) - (k^2)*T(n-2,k).
G.f. for columns: x/(1 - k*x)^2.
E.g.f. for columns: x*exp(k*x).
T(n,1) = A001477(n).
T(n,2) = A001787(n).
T(n,3) = A027471(n+1).
T(n,4) = A002697(n).
T(n,5) = A053464(n).
T(n,6) = A053469(n), n > 0.
T(n,7) = A027473(n), n > 0.
T(n,8) = A053539(n).
T(n,9) = A053540(n), n > 0.
T(n,10) = A053541(n), n > 0.
T(n,11) = A081127(n).
T(n,12) = A081128(n).

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

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

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A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

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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).

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cp's OEIS Frontend

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

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Examples

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Mathematica

Maxima

Formula

A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

Views

Author

Keywords

Comments

Examples

References

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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

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Programs

Mathematica

Maxima

Formula

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).