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.

1, 1, 2, -2, -3, 1, 1, -2, 2, 17, -9, -32, 12, 24, -6, -8, 1, 1, -2, -6, 25, 71, -80, -218, 126, 284, -106, -190, 48, 69, -11, -13, 1, 1, 3, -6, -70, 101, 506, -453, -1592, 980, 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

Offset: 0

Author

Stephen O'Sullivan, Jan 15 2016

Keywords

• sign
• tabf

Comments

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).

Row n is of length 5n+1.

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).)

Examples

			1;
1,2,-2,-3,1,1;
-2,2,17,-9,-32,12,24,-6,-8,1,1;
-2,-6,25,71,-80,-218,126,284,-106,-190,48,69,-11,-13,1,1;
		

Links

• Stephen O'Sullivan, Table of n, a(n) for n = 0..1070
• 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.

Crossrefs

Cf. A267120, A267482, A267483, A267484, A267486.

Programs

Maple

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);

Mathematica

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 *)

Formula

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}.

Extensions

Added row length