A219234 -id:A219234 - cp's OEIS frontend

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.

0, 1, 0, 3, 4, 1, 0, 9, 24, 22, 8, 1, 0, 27, 108, 171, 136, 57, 12, 1, 0, 81, 432, 972, 1200, 886, 400, 108, 16, 1, 0, 243, 1620, 4725, 7920, 8430, 5944, 2810, 880, 175, 20, 1, 0, 729, 5832, 20898, 44280, 61695, 59472, 40636, 19824, 6855, 1640, 258, 24, 1

Offset: 0

Franck Maminirina Ramaharo, Feb 26 2018

T(n,k) is the number of state diagrams having k components of n connected summed trefoil knots.

Row sums gives A001018.

			The triangle T(n, k) begins:
n\k 0     1      2      3       4       5       6      7        8       9
0:  0     1
1:  0     3      4      1
2:  0     9     24     22       8       1
3:  0    27    108    171     136      57      12       1
4:  0    81    432    972    1200     886     400     108      16       1

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.

Michael De Vlieger, Table of n, a(n) for n = 0..10301 (rows 0 <= n <= 100, flattened.)
Ryo Hanaki, Pseudo diagrams of knots, links and spatial graphs, Osaka Journal of Mathematics, Vol. 47 (2010), 863-883.
Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
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 bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Row 2: row 5 of A158454.
Row 3: row 2 of A220665.
Row 4: row 5 of A219234.

row[n_] := CoefficientList[x*(x^2 + 4*x + 3)^n, x]; Array[row, 7, 0] // Flatten (* Jean-François Alcover, Mar 16 2018 *)

g(x, y) := taylor(x/(1 - y*(x^2 + 4*x + 3)), y, 0, 10)$
a : makelist(ratcoef(g(x, y), y, n), n, 0, 10)$
T : []$
for i:1 thru 11 do
  T : append(T, makelist(ratcoef(a[i], x, n), n, 0, 2*i - 1))$
T;

T(n, k) = polcoeff(x*(x^2 + 4*x + 3)^n, k);
tabf(nn) = for (n=0, nn, for (k=0, 2*n+1, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 03 2018

T(n,k) = coefficients of x*(x^2 + 4*x + 3)^n.
T(n,k) = T(n-1,k-2) + 4*T(n-1,k-1) + 3*T(n-1,k), with T(n,0) = 0, T(n,1) = 3^n and  T(n,2) =  4*n*3^(n-1).
T(n,n+k+1) = A152842(2*n,n+k) and T(n,n-k) = A152842(2*n,n+k+1), for n >= k >= 0.
T(n,1) =  A000244(n).
T(n,2) =  A120908(n).
T(n,n+1) = A069835(n).
T(n,2*n-1) = A139272(n).
T(n,2*n) = A008586(n).
T(n,2*n-2) = A140138(4*n) = A185872(2n,2) for n >= 1.
G.f.: x/(1 - y*(x^2 + 4*x + 3)).

Typo in row 6 corrected by Jean-François Alcover, Mar 16 2018

A219240 Coefficient array for the cube of Chebyshev's S polynomials.

Original entry on oeis.org

1, 0, 0, 0, 1, -1, 0, 3, 0, -3, 0, 1, 0, 0, 0, -8, 0, 12, 0, -6, 0, 1, 1, 0, -9, 0, 30, 0, -45, 0, 30, 0, -9, 0, 1, 0, 0, 0, 27, 0, -108, 0, 171, 0, -136, 0, 57, 0, -12, 0, 1, -1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1, 0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1

Offset: 0

Wolfdieter Lang, Dec 12 2012

The row lengths sequence is 3*n+1 = A016777(n).
For the coefficient triangle for Chebyshev's S polynomials see A049310.
The o.g.f. for S(n,x)^3, n >= 0, is GS(3;x,z) = (1+z^2+2*z*x)/ ((1+z^2-z*x)*(1+z^2-z*x*(x^2-3))). This is obtained from the de Moivre-Binet formula for S(n,x) and the binomial theorem.
In general the monic integer Chebyshev polynomial tau(n,x):= R(2*n+1,x)/x enters, where R(n,x) = 2*T(n,x/2) with Chebyshev's T polynomial (for R see A127672), and the coefficient triangle for tau is given in A111125 (here for the third power of S only tau(0,x) = 1 and tau(1,x) = x^2 - 3 enter).

			The array a(n,m) begins:
n\m   0  1  2  3  4    5   6    7  8    9 10  11 12  13 14 15
n=0:  1
n=1:  0  0  0  1
n=2: -1  0  3  0 -3    0   1
n=3:  0  0  0 -8  0   12   0  -6   0    1
n=4:  1  0 -9  0 30    0 -45   0  30    0 -9   0  1
n=5:  0  0  0 27  0 -108   0  171  0 -136  0  57  0 -12  0  1
...
Row n=6: [-1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1],
Row n=7: [0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1],
Row n=8: [1, 0, -30, 0, 345, 0, -1921, 0, 5598, 0, -9540, 0, 10212, 0, -7137, 0, 3303, 0, -1003, 0, 192, 0, -21, 0, 1].
n=2: S(2,x)^3 = (x^2 - 1)^3 = -1 + 3*x^2 - 3*x^4 + x^6.
n=3: S(3,x)^3 = (x^3 - 2*x)^3 = -8*x^3 + 12*x^5 - 6*x^7 + x^9.

Cf. A049310, A127672, A158454 (square of S polynomials), A219234 (fourth power of S polynomials).

a(n,m) = [x^m] S(n, x)^3, n >= 0, 0 <= m <= 3*n, with Chebyshev's S polynomials (see A049310).
a(n,m) = [x^m]([z^n] GS(3;x,z)), with the o.g.f. GS(3;x,z) given above in a comment.
The row polynomials p(n, x) := Sum_{m=0..3*n} a(n,m)*x^m = S(n, x)^3 are (S(3*n+2, x) - 3*S(n, x))/(x^2 - 4). For the factorization of S polynomials see comments on A049310. - Wolfdieter Lang, Apr 09 2018

cp's OEIS Frontend

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.

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A219240 Coefficient array for the cube of Chebyshev's S polynomials.

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cp's OEIS Frontend

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

Extensions

A219240 Coefficient array for the cube of Chebyshev's S polynomials.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.