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A267485 Triangle of coefficients of Gaussian polynomials [2n+5,5]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=5n.

%I A267485 #21 Dec 19 2017 01:46:34
%S A267485 1,1,2,-2,-3,1,1,-2,2,17,-9,-32,12,24,-6,-8,1,1,-2,-6,25,71,-80,-218,
%T A267485 126,284,-106,-190,48,69,-11,-13,1,1,3,-6,-70,101,506,-453,-1592,980,
%U A267485 2658,-1201,-2608,886,1581,-400,-600,108,139,-16,-18,1,1,3,12,-88,-334,779,2774,-3226,-10389,7709,21620,-11608,-27865,11496,23591,-7645,-13512,3427,5276,-1020,-1385,193,234,-21,-23,1,1
%N A267485 Triangle of coefficients of Gaussian polynomials [2n+5,5]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=5n.
%C A267485 The entry a(n,k), n >= 0, k = 0,1,...,g, where g=5n, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+5,5]_q = Sum_{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
%C A267485 Row n is of length 5n+1.
%C A267485 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 A267485 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 A267485 Stephen O'Sullivan, <a href="/A267485/b267485.txt">Table of n, a(n) for n = 0..1070</a>
%H A267485 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 A267485 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient">Gaussian binomial coefficients</a>.
%F A267485 G.f. for row polynomial: G(n,x) = (d^5/dt^5)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/5!)|_{t=0}.
%e A267485 1;
%e A267485 1,2,-2,-3,1,1;
%e A267485 -2,2,17,-9,-32,12,24,-6,-8,1,1;
%e A267485 -2,-6,25,71,-80,-218,126,284,-106,-190,48,69,-11,-13,1,1;
%p A267485 A267485 := 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+2),t$5)/5!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267485(n, k), k = 0 .. 5*n), n = 0 .. 5);
%t A267485 row[n_] := 1/5! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 5}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* From A267120 entry by _Jean-François Alcover_ *)
%Y A267485 Cf. A267120, A267482, A267483, A267484, A267486.
%K A267485 sign,tabf
%O A267485 0,3
%A A267485 _Stephen O'Sullivan_, Jan 15 2016
%E A267485 Added row length