This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A219234 #12 Apr 17 2018 05:28:58
%S A219234 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 A219234 1,0,0,81,-432,972,-1200,886,-400,108,-16,1,1,-24,236,-1228,3678,
%U A219234 -6612,7490,-5532,2701,-864,174,-20,1,0,0,256,-2560,11136,-27776,44176,-47232,34912,-18048,6504,-1600,256,-24,1
%N A219234 Coefficient array for the fourth power of Chebyshev's S-polynomials as a function of x^2.
%C A219234 The row lengths sequence for this array is 2*n+1, given in A005408.
%C A219234 The coefficient triangle for the monic Chebyshev S-polynomials S(n,x) = U(n,x/2) are given in A049310.
%C A219234 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 A219234 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 A219234 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 A219234 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 A219234 The irregular triangle a(n, m) starts:
%e A219234 n\m 0 1 2 3 4 5 6 7 8 9 10 11 12
%e A219234 0: 1
%e A219234 1: 0 0 1
%e A219234 2: 1 -4 6 -4 1
%e A219234 3: 0 0 16 -32 24 -8 1
%e A219234 4: 1 -12 58 -144 195 -144 58 -12 1
%e A219234 5: 0 0 81 -432 972 -1200 886 -400 108 -16 1
%e A219234 6: 1 -24 236 -1228 3678 -6612 7490 -5532 2701 -864 174 -20 1
%e A219234 ...
%e A219234 Row n=7: [0, 0, 256, -2560, 11136, -27776, 44176, -47232, 34912, -18048, 6504, -1600, 256, -24, 1].
%e A219234 Row n=8: [1, -40, 660, -5828, 30194, -96780, 203374, -293464, 300231, -222112, 119938, -47244, 13415, -2672, 354, -28, 1].
%e A219234 Row n=1 polynomial p(1,x) = 1*x^2 = S(1,sqrt(x))^4 = (sqrt(x))^4.
%e A219234 Row n=2 polynomial p(2,x) = 1 - 4*x + 6*x^2 - 4*x^3 + 1*x^4 =
%e A219234 S(2,sqrt(x))^4 = (-1+x)^4.
%Y A219234 Cf. A049310, A127672, A158454, A181878, A219240.
%K A219234 sign,easy,tabf
%O A219234 0,6
%A A219234 _Wolfdieter Lang_, Nov 28 2012