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A277429 Numerators of the Fabius function F(3/2^n).

%I A277429 #8 Oct 16 2016 14:15:13
%S A277429 67,73,46657,25219,29407171,10997359,109661317247,31733679209,
%T A277429 558462830097043,132566737763827,646476041042787542443,
%U A277429 130499244418507180561,2411172049639892707896547,424191560077453917728503,84883189962706557116984038531,172244289373664036915914887721
%N A277429 Numerators of the Fabius function F(3/2^n).
%C A277429 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 A277429 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 A277429 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 A277429 G. A. Edgar, <a href="http://people.math.osu.edu/edgar.2/selfdiff/">Examples of self differential functions</a>
%H A277429 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 A277429 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fabius_function">Fabius function</a>
%e A277429 A277429/A277430 = 67/72, 73/288, 46657/2073600, 25219/33177600, 29407171/2809213747200, ... (starting from n = 2)
%t A277429 c[0] = 1;
%t A277429 c[k_] := c[k] = Sum[((-1)^(k - r) c[r])/(1 + 2 k - 2 r)!, {r, 0, k - 1}]/(4^k - 1);
%t A277429 t[n_] := Mod[2 n + Sum[(-1)^Binomial[n, k], {k, 1, n}], 3];
%t A277429 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 A277429 Table[Numerator[f[3/2^n]], {n, 2, 20}]
%Y A277429 Cf. A277430 (denominators), A272755, A272757, A272343.
%K A277429 nonn,frac
%O A277429 2,1
%A A277429 _Vladimir Reshetnikov_, Oct 14 2016