%I #53 May 12 2020 20:59:27

%S 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,

%T 886,400,108,16,1,0,243,1620,4725,7920,8430,5944,2810,880,175,20,1,0,

%U 729,5832,20898,44280,61695,59472,40636,19824,6855,1640,258,24,1

%N 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.

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

%C Row sums gives A001018.

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

%H Michael De Vlieger, <a href="/A299989/b299989.txt">Table of n, a(n) for n = 0..10301</a> (rows 0 <= n <= 100, flattened.)

%H Ryo Hanaki, <a href="https://projecteuclid.org/euclid.ojm/1285334478">Pseudo diagrams of knots, links and spatial graphs</a>, Osaka Journal of Mathematics, Vol. 47 (2010), 863-883.

%H Louis H. Kauffman, <a href="https://doi.org/10.1016/0040-9383(87)90009-7">State models and the Jones polynomial</a>, Topology, Vol. 26 (1987), 395-407.

%H Carolina Medina, Jorge Ramírez-Alfonsín and Gelasio Salazar, <a href="https://arxiv.org/abs/1710.06470">On the number of unknot diagrams</a>, arXiv:1710.06470 [math.CO], 2017.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10569">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/2002.06672">A bracket polynomial for 2-tangle shadows</a>, arXiv:2002.06672 [math.CO], 2020.

%F T(n,k) = coefficients of x*(x^2 + 4*x + 3)^n.

%F 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).

%F 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.

%F T(n,1) = A000244(n).

%F T(n,2) = A120908(n).

%F T(n,n+1) = A069835(n).

%F T(n,2*n-1) = A139272(n).

%F T(n,2*n) = A008586(n).

%F T(n,2*n-2) = A140138(4*n) = A185872(2n,2) for n >= 1.

%F G.f.: x/(1 - y*(x^2 + 4*x + 3)).

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9

%e 0: 0 1

%e 1: 0 3 4 1

%e 2: 0 9 24 22 8 1

%e 3: 0 27 108 171 136 57 12 1

%e 4: 0 81 432 972 1200 886 400 108 16 1

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

%o (Maxima)

%o g(x, y) := taylor(x/(1 - y*(x^2 + 4*x + 3)), y, 0, 10)$

%o a : makelist(ratcoef(g(x, y), y, n), n, 0, 10)$

%o T : []$

%o for i:1 thru 11 do

%o T : append(T, makelist(ratcoef(a[i], x, n), n, 0, 2*i - 1))$

%o T;

%o (PARI) T(n, k) = polcoeff(x*(x^2 + 4*x + 3)^n, k);

%o tabf(nn) = for (n=0, nn, for (k=0, 2*n+1, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Mar 03 2018

%Y Row 2: row 5 of A158454.

%Y Row 3: row 2 of A220665.

%Y Row 4: row 5 of A219234.

%K tabf,easy,nonn

%O 0,4

%A _Franck Maminirina Ramaharo_, Feb 26 2018

%E Typo in row 6 corrected by _Jean-François Alcover_, Mar 16 2018