A128495 Coefficient table for sums of squares of Chebyshev's S-Polynomials.

1, 1, 1, 2, -1, 1, 2, 3, -3, 1, 3, -3, 8, -5, 1, 3, 6, -16, 17, -7, 1, 4, -6, 30, -45, 30, -9, 1, 4, 10, -50, 103, -98, 47, -11, 1, 5, -10, 80, -211, 269, -183, 68, -13, 1, 5, 15, -120, 399, -651, 588, -308, 93, -15, 1, 6, -15, 175, -707, 1432, -1644, 1136, -481, 122, -17, 1, 6, 21, -245, 1190, -2920, 4132, -3608

Offset: 0

Author

Wolfdieter Lang Apr 04 2007

Keywords

• sign
• easy
• tabl

Comments

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.

The triangle for the coefficients of x^2 in S(n,x)^2 is A158454. - Wolfdieter Lang, Oct 18 2012

Examples

			[1]; [1,1]; [2,-1,1]; [2,3,-3,1]; [3,-3,8,-5,1]; [3,6,-16,17,-7,1]; ...
Row polynomial S(2;4,x)=3-3*x^2+8*x^4-5*x^6+x^8 = sum(S(k,x)^2,k=0..4).
(4+2-T(4+1,x/2)*U(4+1,x/2))/(2*(1-(x/2)^2))= S(2;4,x)
		

Links

• W. Lang, First 15 rows.

Crossrefs

Row sums (signed) look like: A004523. Row sums (unsigned): A128496.

Cf. A128494 =S(1; n, m).

Formula

S(2;n,x):=sum(S(k,x)^2,k=0..n)=sum(a(n,m)*x^(2*m),m=0..n), n>=0.

a(n,m)=[x^m](n+2-T(n+1,x/2)*U(n+1,x/2))/(2*(1-(x/2)^2)).