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A272755 Numerators of the Fabius function F(1/2^n). 7

%I

%S 1,1,5,1,143,19,1153,583,1616353,132809,134926369,46840699,

%T 67545496213157,4068990560161,411124285571171,1204567303451311,

%U 73419800947733963069,4146897304424408411,86773346866163284480799923,18814360006695807527868793,539741515875650532056045666422369

%N Numerators of the Fabius function F(1/2^n).

%C 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.

%D 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.

%H Juan Arias de Reyna, <a href="https://arxiv.org/abs/1702.05442">An infinitely differentiable function with compact support: Definition and properties</a>, arXiv:1702.05442 [math.CA], 2017.

%H Juan Arias de Reyna, <a href="https://arxiv.org/abs/1702.06487">On the arithmetic of Fabius function</a>, arXiv:1702.06487 [math.NT], 2017.

%H Yuri Dimitrov, G. A. Edgar, <a href="http://people.math.osu.edu/edgar.2/preprints/dimitrov/paper.pdf">Solutions of Self-differential Functional Equations</a>

%H G. A. Edgar, <a href="http://people.math.osu.edu/edgar.2/selfdiff/">Examples of self differential functions</a>

%H J. Fabius, <a href="http://dx.doi.org/10.1007/BF00536652">A probabilistic example of a nowhere analytic C^infty-function</a>, Probability Theory and Related Fields, June 1966, Volume 5, Issue 2, pp 173-174.

%H Jan Kristian Haugland, <a href="http://arxiv.org/abs/1609.07999">Evaluating the Fabius function</a>, arXiv:1609.07999 [math.GM], 23 Sep 2016.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fabius_function">Fabius function</a>

%F 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

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

%t c[0] = 1; c[n_] := c[n] = Sum[(-1)^k c[n - k]/(2 k + 1)!, {k, 1, n}] / (4^n - 1); Numerator@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 *)

%Y Cf. A272757 (denominators), A272343.

%K nonn,frac

%O 0,3

%A _Vladimir Reshetnikov_, May 05 2016

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