This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A219240 #13 Apr 10 2018 15:56:34
%S A219240 1,0,0,0,1,-1,0,3,0,-3,0,1,0,0,0,-8,0,12,0,-6,0,1,1,0,-9,0,30,0,-45,0,
%T A219240 30,0,-9,0,1,0,0,0,27,0,-108,0,171,0,-136,0,57,0,-12,0,1,-1,0,18,0,
%U A219240 -123,0,399,0,-651,0,588,0,-308,0,93,0,-15,0,1,0,0,0,-64,0,480,0,-1488,0,2488,0,-2472,0,1524,0,-588,0,138,0,-18,0,1
%N A219240 Coefficient array for the cube of Chebyshev's S polynomials.
%C A219240 The row lengths sequence is 3*n+1 = A016777(n).
%C A219240 For the coefficient triangle for Chebyshev's S polynomials see A049310.
%C A219240 The o.g.f. for S(n,x)^3, n >= 0, is GS(3;x,z) = (1+z^2+2*z*x)/ ((1+z^2-z*x)*(1+z^2-z*x*(x^2-3))). This is obtained from the de Moivre-Binet formula for S(n,x) and the binomial theorem.
%C A219240 In general the monic integer Chebyshev polynomial tau(n,x):= R(2*n+1,x)/x enters, where R(n,x) = 2*T(n,x/2) with Chebyshev's T polynomial (for R see A127672), and the coefficient triangle for tau is given in A111125 (here for the third power of S only tau(0,x) = 1 and tau(1,x) = x^2 - 3 enter).
%F A219240 a(n,m) = [x^m] S(n, x)^3, n >= 0, 0 <= m <= 3*n, with Chebyshev's S polynomials (see A049310).
%F A219240 a(n,m) = [x^m]([z^n] GS(3;x,z)), with the o.g.f. GS(3;x,z) given above in a comment.
%F A219240 The row polynomials p(n, x) := Sum_{m=0..3*n} a(n,m)*x^m = S(n, x)^3 are (S(3*n+2, x) - 3*S(n, x))/(x^2 - 4). For the factorization of S polynomials see comments on A049310. - _Wolfdieter Lang_, Apr 09 2018
%e A219240 The array a(n,m) begins:
%e A219240 n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e A219240 n=0: 1
%e A219240 n=1: 0 0 0 1
%e A219240 n=2: -1 0 3 0 -3 0 1
%e A219240 n=3: 0 0 0 -8 0 12 0 -6 0 1
%e A219240 n=4: 1 0 -9 0 30 0 -45 0 30 0 -9 0 1
%e A219240 n=5: 0 0 0 27 0 -108 0 171 0 -136 0 57 0 -12 0 1
%e A219240 ...
%e A219240 Row n=6: [-1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1],
%e A219240 Row n=7: [0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1],
%e A219240 Row n=8: [1, 0, -30, 0, 345, 0, -1921, 0, 5598, 0, -9540, 0, 10212, 0, -7137, 0, 3303, 0, -1003, 0, 192, 0, -21, 0, 1].
%e A219240 n=2: S(2,x)^3 = (x^2 - 1)^3 = -1 + 3*x^2 - 3*x^4 + x^6.
%e A219240 n=3: S(3,x)^3 = (x^3 - 2*x)^3 = -8*x^3 + 12*x^5 - 6*x^7 + x^9.
%Y A219240 Cf. A049310, A127672, A158454 (square of S polynomials), A219234 (fourth power of S polynomials).
%K A219240 sign,tabf,easy
%O A219240 0,8
%A A219240 _Wolfdieter Lang_, Dec 12 2012