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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.

%I A299989 #53 May 12 2020 20:59:27
%S A299989 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 A299989 886,400,108,16,1,0,243,1620,4725,7920,8430,5944,2810,880,175,20,1,0,
%U A299989 729,5832,20898,44280,61695,59472,40636,19824,6855,1640,258,24,1
%N 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.
%C A299989 T(n,k) is the number of state diagrams having k components of n connected summed trefoil knots.
%C A299989 Row sums gives A001018.
%D A299989 V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
%H A299989 Michael De Vlieger, <a href="/A299989/b299989.txt">Table of n, a(n) for n = 0..10301</a> (rows 0 <= n <= 100, flattened.)
%H A299989 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 A299989 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 A299989 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 A299989 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 A299989 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 A299989 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 A299989 T(n,k) = coefficients of x*(x^2 + 4*x + 3)^n.
%F A299989 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 A299989 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 A299989 T(n,1) =  A000244(n).
%F A299989 T(n,2) =  A120908(n).
%F A299989 T(n,n+1) = A069835(n).
%F A299989 T(n,2*n-1) = A139272(n).
%F A299989 T(n,2*n) = A008586(n).
%F A299989 T(n,2*n-2) = A140138(4*n) = A185872(2n,2) for n >= 1.
%F A299989 G.f.: x/(1 - y*(x^2 + 4*x + 3)).
%e A299989 The triangle T(n, k) begins:
%e A299989 n\k 0     1      2      3       4       5       6      7        8       9
%e A299989 0:  0     1
%e A299989 1:  0     3      4      1
%e A299989 2:  0     9     24     22       8       1
%e A299989 3:  0    27    108    171     136      57      12       1
%e A299989 4:  0    81    432    972    1200     886     400     108      16       1
%t A299989 row[n_] := CoefficientList[x*(x^2 + 4*x + 3)^n, x]; Array[row, 7, 0] // Flatten (* _Jean-François Alcover_, Mar 16 2018 *)
%o A299989 (Maxima)
%o A299989 g(x, y) := taylor(x/(1 - y*(x^2 + 4*x + 3)), y, 0, 10)$
%o A299989 a : makelist(ratcoef(g(x, y), y, n), n, 0, 10)$
%o A299989 T : []$
%o A299989 for i:1 thru 11 do
%o A299989   T : append(T, makelist(ratcoef(a[i], x, n), n, 0, 2*i - 1))$
%o A299989 T;
%o A299989 (PARI) T(n, k) = polcoeff(x*(x^2 + 4*x + 3)^n, k);
%o A299989 tabf(nn) = for (n=0, nn, for (k=0, 2*n+1, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Mar 03 2018
%Y A299989 Row 2: row 5 of A158454.
%Y A299989 Row 3: row 2 of A220665.
%Y A299989 Row 4: row 5 of A219234.
%K A299989 tabf,easy,nonn
%O A299989 0,4
%A A299989 _Franck Maminirina Ramaharo_, Feb 26 2018
%E A299989 Typo in row 6 corrected by _Jean-François Alcover_, Mar 16 2018