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 #12 Apr 17 2018 05:28:58
%S 1,0,0,1,1,-4,6,-4,1,0,0,16,-32,24,-8,1,1,-12,58,-144,195,-144,58,-12,
%T 1,0,0,81,-432,972,-1200,886,-400,108,-16,1,1,-24,236,-1228,3678,
%U -6612,7490,-5532,2701,-864,174,-20,1,0,0,256,-2560,11136,-27776,44176,-47232,34912,-18048,6504,-1600,256,-24,1
%N Coefficient array for the fourth power of Chebyshev's S-polynomials as a function of x^2.
%C The row lengths sequence for this array is 2*n+1, given in A005408.
%C The coefficient triangle for the monic Chebyshev S-polynomials S(n,x) = U(n,x/2) are given in A049310.
%C The coefficients for S(n,x)^2 are given in A158454 and in A181878 (odd numbered rows shifted by one unit to the left).
%F a(n, m) = [x^(2*m)] S(n, x)^4, n >= 0, with the monic Chebyshev S-polynomials given in terms of the U-polynomials in a comment above.
%F The o.g.f. GS4(x, z) := sum((S(n, x)^4)*z^n,n=0..infinity) = ((1+z)/(1-z))*(1 - (2-3*x^2)*z + z^2)/((1-z*(-2+x^2)+z^2)*(1-z*(2-4*x^2+x^4)+z^2)). For the o.g.f. of the row polynomials p(n,x) :=sum(a(n,m)*x^m,m=0..n) take GS4(sqrt(x), z).
%F The row polynomial p(n, x^2) = Sum_{m=0..2*n} a(n, m)*x^(2*m) = (S(n, x))^4 = (R(4*(n+1), x) - 4*R(2*(n+1), x) + 6)/(x^2 - 4)^2, where R are the monic Chebyshev T polynomials with coefficients given in A127672. For factorizations of the S polynomials see comments on A049310. - _Wolfdieter Lang_, Apr 09 2018
%e The irregular triangle a(n, m) starts:
%e n\m 0 1 2 3 4 5 6 7 8 9 10 11 12
%e 0: 1
%e 1: 0 0 1
%e 2: 1 -4 6 -4 1
%e 3: 0 0 16 -32 24 -8 1
%e 4: 1 -12 58 -144 195 -144 58 -12 1
%e 5: 0 0 81 -432 972 -1200 886 -400 108 -16 1
%e 6: 1 -24 236 -1228 3678 -6612 7490 -5532 2701 -864 174 -20 1
%e ...
%e Row n=7: [0, 0, 256, -2560, 11136, -27776, 44176, -47232, 34912, -18048, 6504, -1600, 256, -24, 1].
%e Row n=8: [1, -40, 660, -5828, 30194, -96780, 203374, -293464, 300231, -222112, 119938, -47244, 13415, -2672, 354, -28, 1].
%e Row n=1 polynomial p(1,x) = 1*x^2 = S(1,sqrt(x))^4 = (sqrt(x))^4.
%e Row n=2 polynomial p(2,x) = 1 - 4*x + 6*x^2 - 4*x^3 + 1*x^4 =
%e S(2,sqrt(x))^4 = (-1+x)^4.
%Y Cf. A049310, A127672, A158454, A181878, A219240.
%K sign,easy,tabf
%O 0,6
%A _Wolfdieter Lang_, Nov 28 2012