cp's OEIS Frontend

A220665 Array of coefficients of powers of x^2 for (S(2*n+1,x)/x)^3, with Chebyshev's S polynomials A049310.

%I A220665 #12 Jun 02 2025 08:10:44
%S A220665 1,-8,12,-6,1,27,-108,171,-136,57,-12,1,-64,480,-1488,2488,-2472,1524,
%T A220665 -588,138,-18,1,125,-1500,7575,-21200,36690,-41700,32211,-17184,6330,
%U A220665 -1580,255,-24,1,-216,3780,-28098,117323,-308688,546864,-680474,611019,-402264,195444,-69894,18153,-3328,408,-30,1
%N A220665 Array of coefficients of powers of x^2 for (S(2*n+1,x)/x)^3, with Chebyshev's S polynomials A049310.
%C A220665 The row lengths sequence of this array is 3*n+1 = A016777(n).
%C A220665 For the coefficient array of S(n,x)^3 see A219240. The present array is the odd part of the bisection of that one divided by x^3.
%C A220665 The row polynomials in powers of x^2 are (S(2*n+1,x)/x)^3 = sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0. The o.g.f. for these row polynomials is GS3odd(x,z) = ((z+1)^2 +2*z*(x^2-3))/ (((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2)). This is obtained from the odd part of the bisection of the o.g.f. for A219240.
%F A220665 a(n,m) = [x^m](S(2*n+1,x)/x)^3, n>=0, 0 <= m <= 3*n.
%F A220665 a(n,m) = [x^m]([z^n]GS3odd(x,z)) with GS3odd(x,z) the o.g.f. for the row polynomials in powers of x^2, given in a comment above.
%e A220665 The array a(n,m) begins:
%e A220665 n\m  0    1     2     3      4     5     6    7    8  9
%e A220665 0:   1
%e A220665 1:  -8   12    -6     1
%e A220665 2:  27 -108   171  -136     57   -12     1
%e A220665 3: -64  480 -1488  2488  -2472  1524  -588  138  -18  1
%e A220665 ...
%e A220665 Row n=4: [125 -1500, 7575, -21200, 36690, -41700, 32211, -17184, 6330, -1580, 255, -24, 1],
%e A220665 Row n=5: [-216, 3780, -28098, 117323, -308688, 546864, -680474, 611019, -402264, 195444, -69894, 18153, -3328, 408, -30, 1],
%e A220665 Row n=6: [343, -8232, 84378, -489608, 1809129, -4562292, 8219967, -10918992, 10927077, -8356272, 4923132, -2240256, 784840, -209580, 41853, -6048, 597, -36, 1],
%e A220665 Row n=1: (S(3,x)/x)^3 = -8 + 12*x^2 - 6*x^4 + 1*x^6, with Chebyshev's S polynomial.
%Y A220665 Cf. A219240, A220666 (even part of the bisection).
%K A220665 sign,easy,tabf
%O A220665 0,2
%A A220665 _Wolfdieter Lang_, Dec 17 2012