A267484 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.

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

Offset

0,6

Comments

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).

Row n is of length 4n+3.

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..902

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^4/dt^4)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/4!)|_{t=0}.

Example

-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,-121,52,187,-67,-143,38,58,-10,-12,1,1;
-3,3,64,-37,-357,168,883,-361,-1154,397,875,-239,-399,80,108,-14,-16,1,1;

Maple

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);

Mathematica

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 *)

Crossrefs

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

Sequence in context: A019901 A187059 A267120 * A181697 A317175 A257480

Adjacent sequences: A267481 A267482 A267483 * A267485 A267486 A267487

Keyword

sign,tabf

Author

Stephen O'Sullivan, Jan 15 2016

Status

approved