cp's OEIS Frontend

A219235 Coefficient array for the third power of the monic integer Chebyshev polynomials 2*T(2*n+1,x/2)/x as a function of x^2.

%I A219235 #4 Nov 28 2012 12:32:09
%S A219235 1,-27,27,-9,1,125,-375,450,-275,90,-15,1,-343,2058,-5145,7007,-5733,
%T A219235 2940,-952,189,-21,1,729,-7290,30861,-72927,107406,-104652,69768,
%U A219235 -32319,10395,-2277,324,-27,1,-1331,19965,-127776,461857,-1058145,1641486,-1797818,1427679,-834900,361790,-115830,27027,-4466,495,-33,1
%N A219235 Coefficient array for the third power of the monic integer Chebyshev polynomials 2*T(2*n+1,x/2)/x as a function of x^2.
%C A219235 The length of row n of this array is 3*n+1; see A016777.
%C A219235 Define tau(n,x):= C(2*n+1,x)/x, with C the monic integer Chebyshev T-polynomials with their coefficients given in A127672 (C(n,x) = 2*T(n,x/2) is there called R(n,x)). The coefficients of the x^2-powers of tau(n) are found as signed A111125 (see the last of the comments from Oct 18 2012 there). The irregular triangle a(n,m) appears in tau(n,x)^3 = sum(a(n,m)*x(2*m),m=0..3*n), n>=0.
%C A219235 The o.g.f. of the row polynomials as function of x^2 is G(3;x,z) := sum(tau(n,x)^3*z^n, n=0..infinity) =
%C A219235   (1 - (23-17*x^2+3*x^4)*z*(1-z) - z^3)/(((z+1)^2-x^2*z)*((z+1)^2-z*x^2*(x^2-3)^2)). From the odd part of the bisection of the o.g.f. for C(n,x)^3 divided by x^3.
%F A219235 a(n,m) = [x^(2*m)] tau(n,x)^3, n>=0, m=0,1,...,3*n, with the monic integer polynomials tau(n,x) defined above in a comment.
%e A219235 The array a(n,m) begins:
%e A219235 n\m   0    1     2    3     4     5     6   7   8  9 ...
%e A219235 0:    1
%e A219235 1:  -27   27    -9    1
%e A219235 2:  125 -375   450 -275    90   -15     1
%e A219235 3: -343 2058 -5145 7007 -5733  2940  -952 189 -21  1
%e A219235 ...
%e A219235 Row n=4: [729, -7290, 30861, -72927, 107406, -104652, 69768, -32319, 10395, -2277, 324, -27, 1].
%e A219235 Row n=5: [-1331, 19965, -127776, 461857, -1058145, 1641486, -1797818, 1427679, -834900, 361790, -115830, 27027, -4466, 495, -33, 1].
%e A219235 Row n=1 polynomial p(1,x) := -27 + 27*x - 9*x^2 + 1*x^3 with p(1,x^2) = tau(1,x)^3 = (-3 + x^2)^3 = -27+27*x^2-9*x^4+x^6.
%Y A219235 Cf. A111125 (tau(n,x) coefficients if signed).
%K A219235 sign,tabf
%O A219235 0,2
%A A219235 _Wolfdieter Lang_, Nov 27 2012