A134828 Numerator of moments of Chebyshev U- (or S-) polynomials.

1, 0, 1, 0, 1, 0, 5, 0, 7, 0, 21, 0, 33, 0, 429, 0, 715, 0, 2431, 0, 4199, 0, 29393, 0, 52003, 0, 185725, 0, 334305, 0, 9694845, 0, 17678835, 0, 64822395, 0, 119409675, 0, 883631595, 0, 1641030105, 0, 6116566755, 0, 11435320455, 0, 171529806825, 0

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008

Keywords

• nonn
• easy
• frac

Comments

The denominators are given in A134829.

Essentially the absolute values of numerators in expansion of sqrt(1+x^2). - Arkadiusz Wesolowski, Jan 17 2013

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. A098597 (coincides with numerators for even n).

Programs

Mathematica

f[n_] := Numerator[CatalanNumber[n]/2^n]; Riffle[Array[f, 24, 0], 0] (* Arkadiusz Wesolowski, Jan 17 2013 *)

Formula

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

a(n)=0 if n is odd, a(n) = numerator(C(n/2)/2^n) if n is even, with the Catalan numbers C(n):=A000108(n).