A267483 - OEIS

%I #34 Dec 18 2017 22:48:17

%S 1,1,0,-1,1,1,1,2,-2,-3,1,1,0,-2,4,7,-4,-5,1,1,1,3,-6,-13,11,16,-6,-7,

%T 1,1,0,-3,9,22,-24,-40,22,29,-8,-9,1,1,1,4,-12,-34,46,86,-62,-91,37,

%U 46,-10,-11,1,1,0,-4,16,50,-80,-166,148,239,-128,-174,56,67,-12,-13,1,1,1,5,-20,-70,130,296,-314,-553,367,541,-230,-297,79,92,-14,-15,1,1

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

%C The entry a(n,k), n >= 0, k = 0,1,...,g, where g=2n+1, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+3,2]_q = Sum_{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).

%C Row n is of length 2n+2.

%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 (A053120) 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="/A267483/b267483.txt">Table of n, a(n) for n = 0..991</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^2/dt^2)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2)))/2!|_{t=0}.

%e 1,1;

%e 0,-1,1,1;

%e 1,2,-2,-3,1,1;

%e 0,-2,4,7,-4,-5,1,1;

%e 1,3,-6,-13,11,16,-6,-7,1,1;

%e 0,-3,9,22,-24,-40,22,29,-8,-9,1,1;

%e 1,4,-12,-34,46,86,-62,-91,37,46,-10,-11,1,1;

%e 0,-4,16,50,-80,-166,148,239,-128,-174,56,67,-12,-13,1,1;

%e 1,5,-20,-70,130,296,-314,-553,367,541,-230,-297,79,92,-14,-15,1,1;

%p A267483 := 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$2)/2)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267483(n, k), k = 0 .. 2*n+1), n = 0 .. 20);

%p # More efficient:

%p N:= 20: # to get rows 0 to N

%p P[0]:= (1+t)*(t^2 + t*x + 1):

%p B[0]:= 1+x:

%p for n from 1 to N do

%p   P[n]:= expand(series(P[n-1]*(1+t^2+2*t*orthopoly[T](n+1,x/2)),t,3));

%p   B[n]:= coeff(P[n],t,2);

%p od:

%p seq(seq(coeff(B[n],x,j),j=0..2*n+1),n=0..N); # From A267120 entry by _Robert Israel_

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

%Y Cf. A053120, A267120, A267482, A267484, A267485, A267486.

%K sign,tabf

%O 0,8

%A _Stephen O'Sullivan_, Jan 15 2016