

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*(1x)) 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 AriasdeReyna, 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., Nonclassical methods of the approximation theory in boundary value problems, Naukova Dumka, Kiev (1979) (in Russian), pages 117125.


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 Selfdifferential Functional Equations
G. A. Edgar, Examples of self differential functions
J. Fabius, A probabilistic example of a nowhere analytic C^inftyfunction, 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..n1} (2^(k*(k1)/2)/(nk+1)!)*d(k))/((2^n1)*2^(n*(n1)/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



