OFFSET
0,2
COMMENTS
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).
Row n is of length 6n+4.
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.
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).)
LINKS
Stephen O'Sullivan, Table of n, a(n) for n = 0..1343
S. O'Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, Journal of Computational Physics, 300 (2015), 665-678.
Wikipedia, Gaussian binomial coefficients.
FORMULA
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}.
EXAMPLE
-1,-2,1,1;
0,2,-2,-15,7,17,-5,-7,1,1;
-2,-6,25,71,-80,-218,126,284,-106,-190,48,69,-11,-13,1,1;
MAPLE
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Stephen O'Sullivan, Jan 15 2016
EXTENSIONS
Added row length
STATUS
approved