%I #8 Oct 16 2016 14:15:13

%S 67,73,46657,25219,29407171,10997359,109661317247,31733679209,

%T 558462830097043,132566737763827,646476041042787542443,

%U 130499244418507180561,2411172049639892707896547,424191560077453917728503,84883189962706557116984038531,172244289373664036915914887721

%N Numerators of the Fabius function F(3/2^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 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fabius_function">Fabius function</a>

%e A277429/A277430 = 67/72, 73/288, 46657/2073600, 25219/33177600, 29407171/2809213747200, ... (starting from n = 2)

%t c[0] = 1;

%t c[k_] := c[k] = Sum[((-1)^(k - r) c[r])/(1 + 2 k - 2 r)!, {r, 0, k - 1}]/(4^k - 1);

%t t[n_] := Mod[2 n + Sum[(-1)^Binomial[n, k], {k, 1, n}], 3];

%t f[x_] := Module[{k = Numerator[x], n = Log2[Denominator[x]]}, Sum[((-1)^(q + t[p - 1]) 2^(-(n - 1) n/2) (1/2 - p + k)^(n - 2 q) c[q])/(4^q (n - 2 q)!), {p, 1, k}, {q, 0, n/2}]];

%t Table[Numerator[f[3/2^n]], {n, 2, 20}]

%Y Cf. A277430 (denominators), A272755, A272757, A272343.

%K nonn,frac

%O 2,1

%A _Vladimir Reshetnikov_, Oct 14 2016