A288163 Integers related to the half moments of Rvachëv function.

1, 1, 5, 84, 4004, 494760, 150120600, 107969547840, 179605731622464, 678695382464158080, 5745964983105758544000, 107798142804281290451059200, 4441362930723337358985334172160, 398854836980938754158182857661404160, 77576833096847783279235708819073596288000

Offset: 0

Author

Juan Arias-de-Reyna, Jun 06 2017

Keywords

• nonn

Comments

These numbers determine 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.

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, A288161.

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));
a[n_] := (n + 1)! Product[(2^k - 1), {k, 1, n}] d[n];
Table[a[n], {n, 0, 14}]

Formula

a(n) = (n+1)!*Product_{k=1..n}(2^k-1)*d(n) where d(n) are the rationals defined by the recurrence d(0)=1; d(n)=Sum_{k=0..n-1}[binomial(n+1,k)d(k)]/((n+1)*(2^n-1)) (cf. A288161).