A288161 Denominator of half moments of Rvachëv function.

2, 18, 6, 1350, 270, 23814, 17010, 65063250, 7229250, 9762090030, 4437313650, 8267713725521250, 635977978886250, 81188783595533250, 297692206516955250, 22510683177794610356250, 1564913803803903393750, 40011216302189267004656036250, 10529267447944543948593693750

Offset: 1

Author

Juan Arias-de-Reyna, Jun 06 2017

Keywords

• nonn
• frac

Comments

a(n) is equal to the denominator of the integral over (0,1) of n*t^(n-1)*up(t).

These numbers are the half moments of the Rvachëv function. The Rvachëv function is related to the Fabius function, up(x)=F(x+1) for |x|<1 and up(x)=0 for |x|>=1.

The sequence of numerators is not in the OEIS because it appears t coincide with A272755: Numerators of Fabius function F(1/2^n). In fact d(n) = n! 2^binomial(n,2)F(1/2^n). The coincidence depends on the fact that n! 2^binomial(n,2) divides the denominator of F(1/2^n). It is true that 2^binomial(n,2) divides this denominator, but I do not see any reason for n! to divide this denominator.

Examples

			The rationals d(n) are  1/2, 5/18, 1/6, 143/1350, 19/270,  ...
		

Links

• J. Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 [math.CA], 2017.
• J. Arias de Reyna, Arithmetic of the Fabius function, arXiv:1702.06487 [math.NT], 2017.

Crossrefs

Cf. A287936, A287937, A287938.

Programs

Mathematica

d[0] = 1;
d[n_] := d[n] =
  Sum[Binomial[n + 1, k] d[k], {k, 0, n - 1}]/((n + 1)*(2^n - 1));
Table[Denominator[d[n]], {n, 1, 20}]

Formula

Recurrence d(0)=1; d(n)=Sum_{k=0..n-1}(binomial(n+1,k)d(k))/((n+1)*(2^n-1)) with a(n) are the denominators of d(n).

It may also be defined to be the only sequence d(n) with d(0)=1 and such that the function f(x)=Sum_{n>=0} d(n) x^n/n! satisfies x*f(2x)=(e^x-1)*f(x).