A267484 - OEIS

%I #23 Dec 19 2017 02:09:14

%S -1,1,1,-1,0,5,-2,-4,1,1,-2,2,17,-9,-32,12,24,-6,-8,1,1,-2,0,31,-12,

%T -121,52,187,-67,-143,38,58,-10,-12,1,1,-3,3,64,-37,-357,168,883,-361,

%U -1154,397,875,-239,-399,80,108,-14,-16,1,1,-3,0,94,-36,-808,366,3019,-1312,-6023,2351,7182,-2415,-5439,1512,2686,-587,-863,138,174,-18,-20,1,1,-4,4,158,-94,-1720,856,8611,-3923,-23883,10003,40648,-15328,-45241,14957,34203,-9623,-17893,4135,6485,-1175,-1599,212,256,-22,-24,1,1

%N Triangle of coefficients of Gaussian polynomials [2n+5,4]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=4n+2.

%C The entry a(n,k), n >= 0, k = 0,1,...,g, where g=4n+2, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2*n+5,4]_q = Sum_{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).

%C Row n is of length 4n+3.

%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="/A267484/b267484.txt">Table of n, a(n) for n = 0..902</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^4/dt^4)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/4!)|_{t=0}.

%e -1,1,1;

%e -1,0,5,-2,-4,1,1;

%e -2,2,17,-9,-32,12,24,-6,-8,1,1;

%e -2,0,31,-12,-121,52,187,-67,-143,38,58,-10,-12,1,1;

%e -3,3,64,-37,-357,168,883,-361,-1154,397,875,-239,-399,80,108,-14,-16,1,1;

%p A267484 := 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$4)/4!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267484(n, k), k = 0 .. 4*n+2), n = 0 .. 20);

%t row[n_] := 1/4! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 4}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* From A267120 entry by _Jean-François Alcover_ *)

%Y Cf. A267120, A267482, A267483, A267485, A267486.

%K sign,tabf

%O 0,6

%A _Stephen O'Sullivan_, Jan 15 2016