%I #24 Apr 07 2024 17:11:54

%S 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 16,-80,148,-128,56,-12,1,1,-20,130,-314,367,-230,79,-14,1,25,-200,

%U 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 Coefficient array for square of Chebyshev S-polynomials.

%C For the coefficients of Chebyshev polynomials S(n,x) see A049310.

%C The row length sequence for this array is A109613 = {1,1,3,3,5,5,...}.

%C The row polynomials (in x^2) for even row numbers are

%C S(2*k,x)^2 = Sum_{m=0..2*k} a(2*k,m)*x^(2*m), k >= 0.

%C For odd row numbers the row polynomials (in x^2) are

%C (S(2*k+1,x)^2)/x^2 = Sum_{m=0..2*k} a(2*k+1,m)*x^(2*m), k >= 0.

%C The o.g.f. for the polynomials S(n,x)^2 is

%C 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 The o.g.f. for (S(2*k,sqrt(x))^2 is

%C (1-2(1-x)z+z^2)/((1-z)*(1 - (2-4x+x^2)z + z^2)).

%C The o.g.f. for ((S(2*k+1,sqrt(x))^2)/x is

%C ((1+z)/(1-z))/(1 - (2-4x+x^2)z + z^2).

%C The row sums A011655(n+1) are the same as those for the triangle A158454.

%C 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 Wolfdieter Lang, <a href="/A181878/a181878.pdf">First ten rows with more details and proofs</a>.

%F a(2*k,m) = (-1)^m*Sum_{j=0..k} binomial(2*k+m-1-2*j, 2*m-1), k >= 0.

%F a(2*k+1,m) = (-1)^m*Sum_{j=0..k} binomial(2*k+1+m-2*j, 2*m+1), k >= 0.

%F This derives from the formula for the entries of the Riordan array A158454.

%F For the o.g.f.s see the comment.

%e The irregular triangle a(n,m) begins:

%e n\m 0 1 2 3 4 5 6 7 8 9 10 ...

%e 0: 1

%e 1: 1

%e 2: 1 -2 1

%e 3: 4 -4 1

%e 4: 1 -6 11 -6 1

%e 5: 9 -24 22 -8 1

%e 6: 1 -12 46 -62 37 -10 1

%e 7: 16 -80 148 -128 56 -12 1

%e 8: 1 -20 130 -314 367 -230 79 -14 1

%e 9: 25 -200 610 -920 771 -376 106 -16 1

%e 10: 1 -30 295 -1106 2083 -2232 1444 -574 137 -18 1

%e ... Reformatted and extended by _Wolfdieter Lang_, Nov 24 2012

%t Join[{{1}, {1}}, CoefficientList[Table[ChebyshevU[n, Sqrt[x]/2]^2, {n, 2, 10}], x]] // Flatten (* _Eric W. Weisstein_, Apr 04 2018 *)

%t Join[{{1}, {1}}, CoefficientList[ChebyshevU[Range[2, 10], Sqrt[x]/2]^2, x]] // Flatten (* _Eric W. Weisstein_, Apr 04 2018 *)

%Y Cf. A158454, A129818.

%K sign,easy,tabf

%O 0,4

%A _Wolfdieter Lang_, Dec 22 2010

%E Corrected by _Wolfdieter Lang_, Jan 21 2011