Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 Apr 10 2018 15:56:34
%S 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 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 -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 Coefficient array for the cube of Chebyshev's S polynomials.
%C The row lengths sequence is 3*n+1 = A016777(n).
%C For the coefficient triangle for Chebyshev's S polynomials see A049310.
%C 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 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 a(n,m) = [x^m] S(n, x)^3, n >= 0, 0 <= m <= 3*n, with Chebyshev's S polynomials (see A049310).
%F 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 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 The array a(n,m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e n=0: 1
%e n=1: 0 0 0 1
%e n=2: -1 0 3 0 -3 0 1
%e n=3: 0 0 0 -8 0 12 0 -6 0 1
%e n=4: 1 0 -9 0 30 0 -45 0 30 0 -9 0 1
%e n=5: 0 0 0 27 0 -108 0 171 0 -136 0 57 0 -12 0 1
%e ...
%e Row n=6: [-1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1],
%e 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 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 n=2: S(2,x)^3 = (x^2 - 1)^3 = -1 + 3*x^2 - 3*x^4 + x^6.
%e n=3: S(3,x)^3 = (x^3 - 2*x)^3 = -8*x^3 + 12*x^5 - 6*x^7 + x^9.
%Y Cf. A049310, A127672, A158454 (square of S polynomials), A219234 (fourth power of S polynomials).
%K sign,tabf,easy
%O 0,8
%A _Wolfdieter Lang_, Dec 12 2012