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%I #22 Dec 19 2017 02:14:59
%S -1,-2,1,1,0,2,-2,-15,7,17,-5,-7,1,1,-2,-6,25,71,-80,-218,126,284,
%T -106,-190,48,69,-11,-13,1,1,0,6,-12,-137,196,945,-811,-2745,1602,
%U 4163,-1780,-3711,1193,2059,-493,-722,123,156,-17,-19,1,1,-3,-12,94,358,-952,-3430,4699,15615,-13467,-39946,24494,63168,-29535,-65638,24206,46512,-13652,-22891,5294,7834,-1386,-1831,234,279,-23,-25,1,1
%N Triangle of coefficients of Gaussian polynomials [2n+7,6]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=6n+3.
%C The entry a(n,k), n >= 0, k = 0,1,...,g, where g=6n+3, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+7,6]_q = Sum_{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
%C Row n is of length 6n+4.
%C The sequence arises in the formal derivation of the stability polynomial B(x) = Sum_{i=0..N} d_i T(iM,x) of rank N, and degree L, where T(iM,x) denotes the Chebyshev polynomial of the first kind of degree iM. The coefficients d_i are determined by order conditions on the stability polynomial.
%C Conjecture: More generally, the Gaussian polynomial [2*n+m+1-(m mod 2),m]_q = Sum_{k=0..g(m;n)} a(m;n,k)*(1+q^2)^k*q^(g(m;n)-k), for m >= 0, n >= 0, where g(m;n) = m*n if m is odd and (2*n+1)*m/2 if m is even, and the tabf array entries a(m;n,k) are the coefficients of the g.f. for the row n polynomials G(m;n,x) = (d^m/dt^m)G(m;n,t,x)/m!|_{t=0}, with G(m;n,t,x) = (1+t)*Product_{k=1..n+(m - m (mod 2))/2}(1 + t^2 + 2*t*T(k,x/2) (Chebyshev's T-polynomials). Hence a(m;n,k) = [x^k]G(m;n,x), for k=0..g(m;n). The present entry is the instance m = 2. (Thanks to _Wolfdieter Lang_ for clarifying the text on the general prescription of a(m;n,k).)
%H Stephen O'Sullivan, <a href="/A267486/b267486.txt">Table of n, a(n) for n = 0..1343</a>
%H S. O'Sullivan, <a href="http://dx.doi.org/10.1016/j.jcp.2015.07.050">A class of high-order Runge-Kutta-Chebyshev stability polynomials</a>, Journal of Computational Physics, 300 (2015), 665-678.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient">Gaussian binomial coefficients</a>.
%F G.f. for row polynomial: G(n,x) = (d^6/dt^6)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/6!)|_{t=0}.
%e -1,-2,1,1;
%e 0,2,-2,-15,7,17,-5,-7,1,1;
%e -2,-6,25,71,-80,-218,126,284,-106,-190,48,69,-11,-13,1,1;
%p A267486 := proc (n, k) local y: y := expand(subs(t = 0, diff((1+t)*product(1+t^2+2*t*ChebyshevT(i, x/2), i = 1 .. n+3),t$6)/6!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267486(n, k), k = 0 .. 6*n+3), n = 0 .. 20);
%t row[n_] := 1/6! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 6}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* From A267120 entry by _Jean-François Alcover_ *)
%Y Cf. A267120, A267482, A267483, A267484, A267485.
%K sign,tabf
%O 0,2
%A _Stephen O'Sullivan_, Jan 15 2016
%E Added row length
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