A220665 -id:A220665 - 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

A220666 Array of coefficients of powers of x^2 for S(2*n,x)^3 with Chebyshev's S polynomials A049310.

Original entry on oeis.org

1, -1, 3, -3, 1, 1, -9, 30, -45, 30, -9, 1, -1, 18, -123, 399, -651, 588, -308, 93, -15, 1, 1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1, -1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1

Offset: 0

Wolfdieter Lang, Dec 17 2012

The row lengths sequence of this array is 3*n+1 = A016777(n).
For the coefficient array of S(n,x)^3 see A219240. The present array is the even part of the bisection of that one.
The row polynomials in powers of x^2 are (S(2*n,x))^3 =
  sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0. The o.g.f. for these row polynomials is GS3even(x,z) = ((z+1)^3 + (1+z)*z*x^2*(3*x^2 - 7))/(((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2)). This is obtained from the even part of the bisection of the o.g.f. for A219240.

			The array a(n,m) begins:
n\m  0    1     2     3      4     5     6    7    8  9
0:   1
1:  -1    3    -3     1
2:   1   -9    30   -45     30    -9     1
3:   1   18  -123   399   -651   588  -308   93  -15  1
...
Row n=4: [1, -30, 345, -1921, 5598, -9540, 10212, -7137, 3303, -1003, 192, -21, 1],
Row n=5: [-1, 45, -780, 6609, -29847, 80718, -141482, 168927, -141636, 84766, -36366, 11091, -2346, 327, -27, 1],
Row n=6: [1, -63, 1533, -18333, 118029, -460815, 1184872, -2118207, 2729922, -2598297, 1854177, -999687, 407472, -124680, 28164, -4553, 498, -33, 1].
Row n=2: S(4,x)^3 = 1 - 9*x^2 + 30*x^4 - 45*x^6 + 30*x^8  - 9*x^10 + 1*x^12.

Cf. A219240, A220665 (odd part of the bisection).

a(n,m) = [x^m] S(2*n,x)^3, n>=0, 0 <= m <= 3*n.

a(n,m) = [x^m]([z^n]GS3even(x,z)) with GS3even(x,z) the o.g.f. for the row polynomials in powers of x^2, given in a comment above.

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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A220666 Array of coefficients of powers of x^2 for S(2*n,x)^3 with Chebyshev's S polynomials A049310.

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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

A220666 Array of coefficients of powers of x^2 for S(2*n,x)^3 with Chebyshev's S polynomials A049310.

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.