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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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

Row sums gives A001018.

REFERENCES

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

LINKS

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.

FORMULA

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)).

EXAMPLE

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

MATHEMATICA

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

PROG

(Maxima)

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;

(PARI) 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

CROSSREFS

Row 2: row 5 of A158454.

Row 3: row 2 of A220665.

Row 4: row 5 of A219234.

Sequence in context: A195788 A201930 A176979 * A058022 A215202 A139344

Adjacent sequences:  A299986 A299987 A299988 * A299990 A299991 A299992

KEYWORD

tabf,easy,nonn

AUTHOR

Franck Maminirina Ramaharo, Feb 26 2018

EXTENSIONS

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

STATUS

approved

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Last modified June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)