This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A277471 #20 Feb 28 2017 23:49:07
%S A277471 2,1,5,105,7007,1298745,619247475,723733375365,2006532782969715,
%T A277471 12889909959143502285,188494585656727188486375,
%U A277471 6188497678605937441851529425,451101946262511157576785806552415,72341127537387548941434093006996374625,25326487488712595887856341442148826764706875
%N A277471 Normalized values of the Fabius function: 2^binomial(n-1, 2) * (2*n)! * A005329(n) * F(1/2^n).
%C A277471 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.
%C A277471 Comment from _Vladimir Reshetnikov_, Jan 25 2017: I just realized that I do not have a rigorous proof that all terms are integers. Could somebody suggest a proof? I would also be very interested to learn the asymptotics of this sequence.
%C A277471 Juan Arias de Reyna proved that all terms are indeed integers. - _Vladimir Reshetnikov_, Feb 28 2017
%D A277471 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 A277471 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 A277471 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 A277471 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 A277471 G. A. Edgar, <a href="http://people.math.osu.edu/edgar.2/selfdiff/">Examples of self differential functions</a>
%H A277471 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 A277471 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 A277471 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fabius_function">Fabius function</a>
%F A277471 a(n) = 2^binomial(n-1, 2) * (2*n)! * A005329(n) * A272755(n) / A272757(n).
%t A277471 c[0] = 1; c[n_] := c[n] = Sum[(-1)^k c[n - k]/(2 k + 1)!, {k, 1, n}] / (4^n - 1); Table[2^(1 - 2 n) (2 n)! QFactorial[n, 2] Sum[c[k] (-1)^k/(n - 2 k)!, {k, 0, n/2}], {n, 0, 15}]
%Y A277471 Cf. A005329, A272755, A272757.
%K A277471 nonn
%O A277471 0,1
%A A277471 _Vladimir Reshetnikov_, Oct 16 2016