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 A181878 #29 May 11 2025 08:57:35
%S A181878 1,1,1,-2,1,4,-4,1,1,-6,11,-6,1,9,-24,22,-8,1,1,-12,46,-62,37,-10,1,
%T A181878 16,-80,148,-128,56,-12,1,1,-20,130,-314,367,-230,79,-14,1,25,-200,
%U A181878 610,-920,771,-376,106,-16,1,1,-30,295,-1106,2083,-2232,1444,-574,137,-18,1,36,-420,1897,-4352,5776,-4744,2486,-832,172,-20,1,1,-42,581,-3108,8518,-13672,13820,-9142,4013,-1158,211,-22,1
%N A181878 Coefficient array for square of Chebyshev S-polynomials.
%C A181878 For the coefficients of Chebyshev polynomials S(n,x) see A049310.
%C A181878 The row length sequence for this array is A109613 = {1,1,3,3,5,5,...}.
%C A181878 The row polynomials (in x^2) for even row numbers are
%C A181878 S(2*k,x)^2 = Sum_{m=0..2*k} a(2*k,m)*x^(2*m), k >= 0.
%C A181878 For odd row numbers the row polynomials (in x^2) are
%C A181878 (S(2*k+1,x)^2)/x^2 = Sum_{m=0..2*k} a(2*k+1,m)*x^(2*m), k >= 0.
%C A181878 The o.g.f. for the polynomials S(n,x)^2 is
%C A181878 S(x,z):=((1+z)/(1-z))/(1 + (2-x^2)z +z^2). See the link for a proof. Therefore the coefficients constitute the Riordan array (1/(1-x^2),x/(1+x)^2) found as A158454.
%C A181878 The o.g.f. for S(2*k,sqrt(x))^2 is
%C A181878 (1-2*(1-x)*z+z^2)/((1-z)*(1 - (2-4*x+x^2)*z + z^2)).
%C A181878 The o.g.f. for (S(2*k+1,sqrt(x))^2)/x is
%C A181878 ((1+z)/(1-z))/(1 - (2-4*x+x^2)*z + z^2).
%C A181878 The row sums A011655(n+1) are the same as those for the triangle A158454.
%C A181878 The alternating row sums for even numbered rows (-1)^n*A007598(n+1) coincide with those of triangle A158454. For odd row numbers n=2k+1 these sums are A049684(k+1), k >= 0 (squares of even-indexed Fibonacci numbers).
%H A181878 Wolfdieter Lang, <a href="/A181878/a181878.pdf">First ten rows with more details and proofs</a>.
%F A181878 a(2*k,m) = (-1)^m*Sum_{j=0..k} binomial(2*k+m-1-2*j, 2*m-1), k >= 0.
%F A181878 a(2*k+1,m) = (-1)^m*Sum_{j=0..k} binomial(2*k+1+m-2*j, 2*m+1), k >= 0.
%F A181878 This derives from the formula for the entries of the Riordan array A158454.
%F A181878 For the o.g.f.s see the comment.
%e A181878 The irregular triangle a(n,m) begins:
%e A181878 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A181878 0: 1
%e A181878 1: 1
%e A181878 2: 1 -2 1
%e A181878 3: 4 -4 1
%e A181878 4: 1 -6 11 -6 1
%e A181878 5: 9 -24 22 -8 1
%e A181878 6: 1 -12 46 -62 37 -10 1
%e A181878 7: 16 -80 148 -128 56 -12 1
%e A181878 8: 1 -20 130 -314 367 -230 79 -14 1
%e A181878 9: 25 -200 610 -920 771 -376 106 -16 1
%e A181878 10: 1 -30 295 -1106 2083 -2232 1444 -574 137 -18 1
%e A181878 ... Reformatted and extended by _Wolfdieter Lang_, Nov 24 2012
%t A181878 Join[{{1}, {1}}, CoefficientList[Table[ChebyshevU[n, Sqrt[x]/2]^2, {n, 2, 10}], x]] // Flatten (* _Eric W. Weisstein_, Apr 04 2018 *)
%t A181878 Join[{{1}, {1}}, CoefficientList[ChebyshevU[Range[2, 10], Sqrt[x]/2]^2, x]] // Flatten (* _Eric W. Weisstein_, Apr 04 2018 *)
%Y A181878 Cf. A158454, A129818.
%K A181878 sign,easy,tabf
%O A181878 0,4
%A A181878 _Wolfdieter Lang_, Dec 22 2010
%E A181878 Corrected by _Wolfdieter Lang_, Jan 21 2011