login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A272757 Denominators of the Fabius function F(1/2^n). 6
1, 2, 72, 288, 2073600, 33177600, 561842749440, 179789679820800, 704200217922109440000, 180275255788060016640000, 1246394851358539387238350848000, 6381541638955721662660356341760000, 292214732887898713986916575925267070976000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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)

REFERENCES

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.

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A272755 (numerators), A272343.

Sequence in context: A185120 A217842 A053318 * A187707 A163274 A030993

Adjacent sequences:  A272754 A272755 A272756 * A272758 A272759 A272760

KEYWORD

nonn,frac

AUTHOR

Vladimir Reshetnikov, May 05 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 17:42 EST 2019. Contains 329768 sequences. (Running on oeis4.)