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A267120 Triangle of coefficients of Gaussian polynomials [2n+3,3]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=3n.

%I A267120 #62 Dec 19 2017 02:14:44
%S A267120 1,0,-1,1,1,-1,0,5,-2,-4,1,1,0,2,-2,-15,7,17,-5,-7,1,1,1,0,-15,6,53,
%T A267120 -23,-67,22,38,-8,-10,1,1,0,-3,3,55,-28,-189,81,261,-90,-182,46,68,
%U A267120 -11,-13,1,1,-1,0,30,-12,-229,106,691,-292,-1010,359,817,-229,-387,79,107,-14,-16,1,1
%N A267120 Triangle of coefficients of Gaussian polynomials [2n+3,3]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=3n.
%C A267120 The entry a(n,k), n >= 0, k = 0,1,...,g, where g=3n, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+3,3]_q = Sum_{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
%C A267120 Row n is of length 3n+1.
%C A267120 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 (A053120). The coefficients d_i are determined by order conditions on the stability polynomial.
%C A267120 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 = 3. (Thanks to _Wolfdieter Lang_ for clarifying the text on the general prescription of a(m;n,k).)
%C A267120 From _Robert Israel_, Jan 15 2016: (Start)
%C A267120 a(n,0) = A056594(n).
%C A267120 a(n,1) = (-1)^((n+1)/2) * A142150(n+1).
%C A267120 a(2n,2) = 5*(-1)^(n+1)*A000217(n), a(2n+1,2) =(-1)^n*(n+1).
%C A267120 It appears that Sum_{j=0..k+1} C(k+1,j)*a(n+2*j,k) = 0.
%C A267120 (End)
%H A267120 Stephen O'Sullivan, <a href="/A267120/b267120.txt">Table of n, a(n) for n = 0..1425</a>
%H A267120 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 A267120 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient">Gaussian binomial coefficients</a>.
%F A267120 G.f. for row polynomial: G(n,x) = (d^3/dt^3)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/3!)|_{t=0}.
%e A267120 The irregular triangle a(n, k) begins:
%e A267120 n/k 0  1   2   3   4    5   6   7   8   9  10 11 12
%e A267120 0:  1
%e A267120 1:  0 -1   1   1
%e A267120 2: -1  0   5  -2  -4   1    1
%e A267120 3:  0  2  -2 -15   7   17  -5  -7   1   1
%e A267120 4:  1  0 -15   6  53  -23 -67  22  38  -8 -10  1  1
%e A267120 ...
%e A267120 Row n=5: 0 -3   3  55 -28 -189  81 261 -90 -182  46 68 -11 -13 1 1;
%e A267120 Row n=6: -1  0 30 -12 -229  106 691 -292 -1010 359 817 -229 -387 79 107 -14 -16 1 1.
%e A267120 Row n=7: 0 4 -4 -134 70 896 -416 -2561 1073 3903 -1415 -3529 1057 1991 -467 -709 121 155 -17 -19 1 1.
%e A267120 ... Reformatted and extended. - _Wolfdieter Lang_, Feb 13 2016
%p A267120 A267120 := 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+1),t$3)/3!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267120(n, k), k = 0 .. 3*n), n = 0 .. 20);
%p A267120 # More efficient:
%p A267120 N:= 20: # to get rows 0 to N
%p A267120 P[0]:= (1+t)*(t^2 + t*x + 1):
%p A267120 B[0]:= 1:
%p A267120 for n from 1 to N do
%p A267120   P[n]:= expand(series(P[n-1]*(1+t^2+2*t*orthopoly[T](n+1,x/2)),t,4));
%p A267120   B[n]:= coeff(P[n],t,3);
%p A267120 od:
%p A267120 seq(seq(coeff(B[n],x,j),j=0..3*n),n=0..N); # _Robert Israel_, Jan 15 2016
%t A267120 row[n_] := 1/3! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 3}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, Jan 16 2016 *)
%Y A267120 Cf. A000217. A056594, A142150, A267482, A267483, A267484, A267485, A267486.
%K A267120 sign,tabf
%O A267120 0,8
%A A267120 _Stephen O'Sullivan_, Jan 10 2016