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.

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

0,4

Comments

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.

References

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

Links

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

Formula

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.

Example

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
  ...

Mathematica

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

Prog

(Maxima) 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));

Crossrefs

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.

Sequence in context: A213626 A329687 A356035 * A222128 A088972 A355570

Adjacent sequences: A321124 A321125 A321126 * A321128 A321129 A321130

Keyword

nonn,easy,tabf

Author

Franck Maminirina Ramaharo, Nov 19 2018

Status

approved