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%I #6 Aug 05 2018 08:28:23
%S 0,1,3,3,1,0,7,14,9,2,0,13,37,43,26,8,1,0,19,72,129,141,98,42,10,1,0,
%T 25,119,291,463,504,378,192,63,12,1,0,31,178,553,1156,1716,1848,1452,
%U 825,330,88,14,1,0,37,249,939,2432,4576,6435,6864,5577,3432,1573
%N Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (x^2 + 4*x + 3)*(x + 1)^(2*n) + (x^2 - 1)*(x^2 + 3*x + 3).
%C The triangle is related to the Kauffman bracket polynomial evaluated at the shadow diagram of the two-bridge knot with Conway's notation C(2n,3).
%H Ji-Young Ham and Joongul Lee, <a href="http://dx.doi.org/10.1142/S0218216516500577">An explicit formula for the A-polynomial of the knot with Conway’s notation C(2n,3)</a>, Journal of Knot Theory and Its Ramifications 25 (2016), 1-9.
%H Ryo Hanaki, <a href="http://dx.doi.org/10.1016/j.topol.2015.08.012">On scannable properties of the original knot from a knot shadow</a>, Topology and its Applications 194 (2015), 296-305.
%H Bin Lu and Jianyuan K. Zhong, <a href="https://arxiv.org/abs/math/0606114">The Kauffman Polynomials of 2-bridge Knots</a>, arXiv:math/0606114 [math.GT], 2006.
%F T(n,1) = A016921(n) and T(n,k) = C(2*n+3,k+1) + (C(2*n+1,k)*(2*k - 2*n) + C(4,k)*(2*k - 3))/(k + 1) for k > 1.
%F T(n,2) = A173247(2*n+1) = A300401(2*n,3).
%F T(n,3) = 2*A099721(n) + 3.
%F T(n,4) = A244730(n) - A002412(n) + 1.
%F T(n,k) = A093560(2*n,k) for n > 2 and k > 4.
%F G.f.: (x^2 + 4*x + 3)/(1 - y*(x + 1)^2) + (x^4 + 3*x^3 + 2*x^2 - 3*x - 3)/(1 - y).
%e The triangle T(n,k) begins:
%e n\k| 0 1 2 3 4 5 9 7 8 9 10 11 12 13 14
%e -------------------------------------------------------------------------------
%e 0 | 0 1 3 3 1
%e 1 | 0 7 14 9 2
%e 2 | 0 13 37 43 26 8 1
%e 3 | 0 19 72 129 141 98 42 10 1
%e 4 | 0 25 119 291 463 504 378 192 63 12 1
%e 5 | 0 31 178 553 1156 1716 1848 1452 825 330 88 14 1
%e 6 | 0 37 249 939 2432 4576 6435 6864 5577 3432 1573 520 117 16 1
%e ...
%p T := proc (n, k) if k = 1 then 6*n + 1 else binomial(2*n + 3, k + 1) + (binomial(2*n + 1, k)*(2*k - 2*n) + binomial(4, k)*(2*k - 3))/(k + 1) end if end proc:
%p for n from 0 to 12 do seq(T(n, k), k = 0 .. max(4, 2*(n + 1))) od;
%t row[n_] := CoefficientList[(x^2 + 4*x + 3)*(x + 1)^(2*n) + (x^2 - 1)*(x^2 + 3*x + 3), x];
%t Array[row, 12, 0] // Flatten
%o (Maxima)
%o T(n, k) := binomial(2*n + 3, k + 1) + (binomial(2*n + 1, k)*(2*k - 2*n) + binomial(4, k)*(2*k - 3))/(k + 1) - kron_delta(1, k)$
%o for n:0 thru 12 do print(makelist(T(n, k), k, 0, max(4, 2*(n + 1))));
%Y Row sums: 4*A004171.
%Y Cf. A093560, A137396, A299989, A300184, A300192, A300453, A300454, A316659.
%K nonn,tabf
%O 0,3
%A _Franck Maminirina Ramaharo_, Jul 18 2018
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