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A277471
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Normalized values of the Fabius function: 2^binomial(n-1, 2) * (2*n)! * A005329(n) * F(1/2^n).
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0
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2, 1, 5, 105, 7007, 1298745, 619247475, 723733375365, 2006532782969715, 12889909959143502285, 188494585656727188486375, 6188497678605937441851529425, 451101946262511157576785806552415, 72341127537387548941434093006996374625, 25326487488712595887856341442148826764706875
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OFFSET
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0,1
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COMMENTS
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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.
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.
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REFERENCES
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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.
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LINKS
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FORMULA
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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