A288161 -id:A288161 - cp's OEIS frontend

A272757 Denominators of the Fabius function F(1/2^n).

1, 2, 72, 288, 2073600, 33177600, 561842749440, 179789679820800, 704200217922109440000, 180275255788060016640000, 1246394851358539387238350848000, 6381541638955721662660356341760000, 292214732887898713986916575925267070976000000

Offset: 0

Vladimir Reshetnikov, May 05 2016

The Fabius function F(x) is the smooth monotone increasing function on [0, 1] satisfying F(0) = 0, F(1) = 1, F'(x) = 2*F(2*x) for 0 < x < 1/2, F'(x) = 2*F(2*(1-x)) for 1/2 < x < 1. It is infinitely differentiable at every point in the interval, but is nowhere analytic. It assumes rational values at dyadic rationals.
From Juan Arias-de-Reyna, Jun 08 2017: (Start)
It is true that n! divides a(n) for all n? This is true for the first 200 terms.
If this is true A272755, the sequence of numerators of F(2^(-n)) is also the sequence of numerators of the half moments of Rvachëv function. (Cf. A288161). (End)

			A272755/A272757 = 1/1, 1/2, 5/72, 1/288, 143/2073600, 19/33177600, 1153/561842749440, 583/179789679820800, ...

Rvachev V. L., Rvachev V. A., Non-classical methods of the approximation theory in boundary value problems, Naukova Dumka, Kiev (1979) (in Russian), pages 117-125.

Vincenzo Librandi, Table of n, a(n) for n = 0..64
Juan Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 [math.CA], 2017.
Juan Arias de Reyna, On the arithmetic of Fabius function, arXiv:1702.06487 [math.NT], 2017.
Yuri Dimitrov, G. A. Edgar, Solutions of Self-differential Functional Equations
G. A. Edgar, Examples of self differential functions
J. Fabius, A probabilistic example of a nowhere analytic C^infty-function, Z. Wahrscheinlichkeitstheorie verw. Gebiete (1966) 5: 173.
Jan Kristian Haugland, Evaluating the Fabius function, arXiv:1609.07999 [math.GM], 23 Sep 2016.
Wikipedia, Fabius function

Cf. A272755 (numerators), A272343.

c[0] = 1; c[n_] := c[n] = Sum[(-1)^k c[n - k]/(2 k + 1)!, {k, 1, n}] / (4^n - 1); Denominator@Table[Sum[c[k] (-1)^k / (n - 2 k)!, {k, 0, n/2}] / 2^((n + 1) n/2), {n, 0, 15}] (* Vladimir Reshetnikov, Oct 16 2016 *)

Recurrence: d(0) = 1, d(n) = (1/(n+1)! + Sum_{k=1..n-1} (2^(k*(k-1)/2)/(n-k+1)!)*d(k))/((2^n-1)*2^(n*(n-1)/2)), where d(n) = A272755(n)/A272757(n). - Vladimir Reshetnikov, Feb 27 2017

A288163 Integers related to the half moments of Rvachëv function.

Original entry on oeis.org

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

Offset: 0

Juan Arias-de-Reyna, Jun 06 2017

nonn

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.

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.

Cf. A287936, A287937, A287938, A288161.

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}]

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).

cp's OEIS Frontend

A272757 Denominators of the Fabius function F(1/2^n).

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