A134829 Denominator of moments of Chebyshev U- (or S-) polynomials.

1, 1, 4, 1, 8, 1, 64, 1, 128, 1, 512, 1, 1024, 1, 16384, 1, 32768, 1, 131072, 1, 262144, 1, 2097152, 1, 4194304, 1, 16777216, 1, 33554432, 1, 1073741824, 1, 2147483648, 1, 8589934592, 1, 17179869184, 1, 137438953472, 1, 274877906944, 1, 1099511627776

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008

Keywords

• nonn
• easy
• frac

Comments

The numerators are given in A134828.

The weight function for Chebyshev's U-polynomials is w(x) = sqrt(1-x^2)*2/Pi for x in [-1,+1]. For the S-polynomials S(n,x) = U(n,x/2) on [-2,+2] it is sqrt(1-x^2)/Pi. For the coefficient of the S-polynomials see A049310.

Examples

			Rationals: [1, 0, 1/4, 0, 1/8, 0, 5/64, 0, 7/128, 0, 21/512, 0, 33/1024, 0, ...].
		

Links

• W. Lang, Rationals and more.

Crossrefs

Cf. A120777 (coincides with denominators for even n).

Formula

a(n) = denominator(r(n)) with r(n) = Integral_{x=-1..+1} (2/Pi)*sqrt(1-x^2)*x^n dx, n >= 0.

a(n)=1 if n is odd, a(n) = denominator(C(n/2)/2^n) if n is even, with the Catalan numbers C(n):=A000108(n).