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 A128494 #26 Jun 09 2021 22:40:56
%S A128494 1,1,1,0,1,1,0,-1,1,1,1,-1,-2,1,1,1,2,-2,-3,1,1,0,2,4,-3,-4,1,1,0,-2,
%T A128494 4,7,-4,-5,1,1,1,-2,-6,7,11,-5,-6,1,1,1,3,-6,-13,11,16,-6,-7,1,1,0,3,
%U A128494 9,-13,-24,16,22,-7,-8,1,1,0,-3,9,22,-24,-40,22,29,-8,-9,1,1,1,-3,-12,22,46,-40,-62,29,37,-9,-10,1,1,1,4,-12
%N A128494 Coefficient table for sums of Chebyshev's S-Polynomials.
%C A128494 See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.
%C A128494 This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k) = a(2*p+1,2*k) = A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1) = a(2*p+2,2*k+1) = A059260(p+k+1,2*k+1)*(-1)^(p+k), k >= 0.
%H A128494 Wolfdieter Lang, <a href="/A128494/a128494.txt">First 15 rows</a>
%H A128494 Per Alexandersson, Luis Angel González-Serrano, and Egor A. Maximenko, Mario Alberto Moctezuma-Salazar, <a href="https://arxiv.org/abs/1912.12725">Symmetric polynomials in the symplectic alphabet and their expression via Dickson-Zhukovsky variables</a>, arXiv:1912.12725 [math.CO], 2019.
%H A128494 Per Alexandersson, Luis Angel González-Serrano, Egor A. Maximenko, and Mario Alberto Moctezuma-Salazar, <a href="https://doi.org/10.37236/9354">Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables z_j = x_j + x_j^(-1)</a>, The Elec. J. of Combinatorics (2021) Vol. 28, No. 1, #P1.56.
%F A128494 S(1;n,x) = Sum_{k=0..n} S(k,x) = Sum_{m=0..n} a(n,m)*x^m, n >= 0.
%F A128494 a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).
%F A128494 G.f. for column m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).
%F A128494 From _Wolfdieter Lang_, Oct 16 2012: (Start)
%F A128494 a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet - de Moivre formula for S and use of the geometric sum.
%F A128494 a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)
%e A128494 The triangle a(n,m) begins:
%e A128494 n\m 0 1 2 3 4 5 6 7 8 9 10
%e A128494 0: 1
%e A128494 1: 1 1
%e A128494 2: 0 1 1
%e A128494 3: 0 -1 1 1
%e A128494 4: 1 -1 -2 1 1
%e A128494 5: 1 2 -2 -3 1 1
%e A128494 6: 0 2 4 -3 -4 1 1
%e A128494 7: 0 -2 4 7 -4 -5 1 1
%e A128494 8: 1 -2 -6 7 11 -5 -6 1 1
%e A128494 9: 1 3 -6 -13 11 16 -6 -7 1 1
%e A128494 10: 0 3 9 -13 -24 16 22 -7 -8 1 1
%e A128494 ... reformatted by _Wolfdieter Lang_, Oct 16 2012
%e A128494 Row polynomial S(1;4,x) = 1 - x - 2*x^2 + x^3 + x^4 = Sum_{k=0..4} S(k,x).
%e A128494 S(4,y)*S(5,y)/y = 3 - 13*y^2 + 16*y^4 - 7*y^6 + y^8, with y=sqrt(2+x) this becomes S(1;4,x).
%e A128494 From _Wolfdieter Lang_, Oct 16 2012: (Start)
%e A128494 S(1;4,x) = (1 - (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).
%e A128494 S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1 + 3*x + 3*x^2 - 4*x^3 - x^4 + x^5. (End)
%Y A128494 Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).
%Y A128494 Cf. A128495 for S(2; n, x) coefficient table.
%Y A128494 The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.
%Y A128494 For m >= 1 the column sequences (without leading zeros) are of the form a(m, 2*k) = a(m, 2*k+1) = ((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.
%K A128494 sign,tabl,easy
%O A128494 0,13
%A A128494 _Wolfdieter Lang_, Apr 04 2007