A219236 Coefficient array for the third power of the monic integer Chebyshev polynomials 2*T(2*n,x/2) as a function of x^2.
8, -8, 12, -6, 1, 8, -48, 108, -112, 54, -12, 1, -8, 108, -558, 1389, -1782, 1287, -546, 135, -18, 1, 8, -192, 1776, -8032, 19308, -27456, 24752, -14688, 5814, -1520, 252, -24, 1, -8, 300, -4350, 31045, -119370, 277137, -419900, 436050, -319770, 168245, -63756, 17250, -3250, 405, -30, 1
Offset: 0
Examples
The irregular triangle a(n,m) begins: n\m 0 1 2 3 4 5 6 7 8 9 ... 0: 8 1: -8 12 -6 1 2: 8 -48 108 -112 54 -12 1 3: -8 108 -558 1389 -1782 1287 -546 135 -18 1 ... Row n=4: [8, -192, 1776, -8032, 19308, -27456, 24752, -14688, 5814, -1520, 252, -24, 1]. Row n=5: [-8, 300, -4350, 31045, -119370, 277137, -419900, 436050, -319770, 168245, -63756, 17250, -3250, 405, -30, 1]. Row n=1 polynomial p(n,1) = -8 + 12*x - 6*x^2 + 1*x^3 = C(2,sqrt(x))^3 = (-2+x)^3.
Formula
a(n,m) = [x^(2*m)] C(2*n,x)^3, with the C-polynomials defined from Chebyshev's T-polynomials in a comment above.
Comments