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 A219240 Coefficient array for the cube of Chebyshev's S polynomials. 3
 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, 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, -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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The row lengths sequence is 3*n+1 = A016777(n). For the coefficient triangle for Chebyshev's S polynomials see A049310. 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. 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). LINKS FORMULA a(n,m) = [x^m] S(n, x)^3, n >= 0, 0 <= m <= 3*n, with Chebyshev's S polynomials (see A049310). 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. 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 EXAMPLE The array a(n,m) begins: n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n=0: 1 n=1: 0 0 0 1 n=2: -1 0 3 0 -3 0 1 n=3: 0 0 0 -8 0 12 0 -6 0 1 n=4: 1 0 -9 0 30 0 -45 0 30 0 -9 0 1 n=5: 0 0 0 27 0 -108 0 171 0 -136 0 57 0 -12 0 1 ... Row n=6: [-1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1], 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], 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]. n=2: S(2,x)^3 = (x^2 - 1)^3 = -1 + 3*x^2 - 3*x^4 + x^6. n=3: S(3,x)^3 = (x^3 - 2*x)^3 = -8*x^3 + 12*x^5 - 6*x^7 + x^9. CROSSREFS Cf. A049310, A127672, A158454 (square of S polynomials), A219234 (fourth power of S polynomials). Sequence in context: A340555 A094901 A030220 * A349612 A277080 A055240 Adjacent sequences: A219237 A219238 A219239 * A219241 A219242 A219243 KEYWORD sign,tabf,easy AUTHOR Wolfdieter Lang, Dec 12 2012 STATUS approved

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Last modified February 1 20:55 EST 2023. Contains 359996 sequences. (Running on oeis4.)