A272755 -id:A272755 - cp's OEIS frontend

A272757 Denominators of the Fabius function F(1/2^n).

1, 2, 72, 288, 2073600, 33177600, 561842749440, 179789679820800, 704200217922109440000, 180275255788060016640000, 1246394851358539387238350848000, 6381541638955721662660356341760000, 292214732887898713986916575925267070976000000

Offset: 0

Vladimir Reshetnikov, May 05 2016

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.
From Juan Arias-de-Reyna, Jun 08 2017: (Start)
It is true that n! divides a(n) for all n? This is true for the first 200 terms.
If this is true A272755, the sequence of numerators of F(2^(-n)) is also the sequence of numerators of the half moments of Rvachëv function. (Cf. A288161). (End)

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

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..64
Juan Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 [math.CA], 2017.
Juan Arias de Reyna, On the arithmetic of Fabius function, arXiv:1702.06487 [math.NT], 2017.
Yuri Dimitrov, G. A. Edgar, Solutions of Self-differential Functional Equations
G. A. Edgar, Examples of self differential functions
J. Fabius, A probabilistic example of a nowhere analytic C^infty-function, Z. Wahrscheinlichkeitstheorie verw. Gebiete (1966) 5: 173.
Jan Kristian Haugland, Evaluating the Fabius function, arXiv:1609.07999 [math.GM], 23 Sep 2016.
Wikipedia, Fabius function

Cf. A272755 (numerators), A272343.

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

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

A287938 Integers associated with moments of Rvachëv function.

Original entry on oeis.org

1, 1, 19, 2915, 2788989, 14754820185, 402830065455939, 54259734183964303995, 34931036957548128175343565, 104968042559556881090071537121985, 1445701512369903326110289606343988638195, 89942525814858602265845303890518923811304544595, 24979493321562411847493262443987087581059026281953954525

Offset: 0

Juan Arias-de-Reyna, Jun 03 2017

nonn

a(n) is equal to the product of (2*n+1)!!*(Product_{k=1..n} (2^(2*k)-1)) and A287936(n)/A287937(n), the moment of the Rvachëv function. The Rvachëv function is related to the Fabius function; up(x)=F(x+1) for |x|<1 and up(x)=0 for |x|>=1, where F is the Fabius function.

J. Arias de Reyna, An infinitely differentiable function with compact support:Definition and properties, arXiv:1702.05442 [math.CA], 2017.
J. Arias de Reyna, Arithmetic of the Fabius function, arXiv:1702.06487 [math.NT], 2017.

Cf. A272755, A272757, A287936, A287937.

c[0] = 1;
c[n_] := c[n] =
   Sum[Binomial[2 n + 1, 2 k] c[k], {k, 0, n - 1}]/((2 n + 1) (2^(2 n) - 1));
a[n_] := a[n] = c[n] (2 n + 1)!! Product[(2^(2 k) - 1), {k, 1, n}];
Table[a[n], {n, 0, 30}]
Table[(-1)^n 4^(-n) (2 n)! (2 n + 1)!! Sum[QBinomial[n, k, 1/4] 2^(-k (3 k + 1)/2)/(2 n + k + 1)! Sum[(-1)^ThueMorse[m] (2 m + 1)^(2 n + k + 1), {m, 0, 2^k - 1}], {k, 0, n}], {n, 0, 12}] (* Vladimir Reshetnikov, Jul 08 2018 *)

a(n) = (2*n+1)!!*(Product_{k=1..n} (2^(2*k)-1))*A287936(n)/A287937(n).

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

Original entry on oeis.org

67, 73, 46657, 25219, 29407171, 10997359, 109661317247, 31733679209, 558462830097043, 132566737763827, 646476041042787542443, 130499244418507180561, 2411172049639892707896547, 424191560077453917728503, 84883189962706557116984038531, 172244289373664036915914887721

Offset: 2

Vladimir Reshetnikov, Oct 14 2016

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.

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

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.

Yuri Dimitrov, G. A. Edgar, Solutions of Self-differential Functional Equations
G. A. Edgar, Examples of self differential functions
J. Fabius, A probabilistic example of a nowhere analytic C^infty-function, Probability Theory and Related Fields, June 1966, Volume 5, Issue 2, pp 173-174.
Wikipedia, Fabius function

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

c[0] = 1;
c[k_] := c[k] = Sum[((-1)^(k - r) c[r])/(1 + 2 k - 2 r)!, {r, 0, k - 1}]/(4^k - 1);
t[n_] := Mod[2 n + Sum[(-1)^Binomial[n, k], {k, 1, n}], 3];
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}]];
Table[Numerator[f[3/2^n]], {n, 2, 20}]

A277430 Denominators of the Fabius function F(3/2^n).

Original entry on oeis.org

72, 288, 2073600, 33177600, 2809213747200, 179789679820800, 704200217922109440000, 180275255788060016640000, 6231974256792696936191754240000, 6381541638955721662660356341760000, 292214732887898713986916575925267070976000000, 1196911545908833132490410294989893922717696000000

Offset: 2

Vladimir Reshetnikov, Oct 14 2016

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.

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

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.

Yuri Dimitrov, G. A. Edgar, Solutions of Self-differential Functional Equations
G. A. Edgar, Examples of self differential functions
J. Fabius, A probabilistic example of a nowhere analytic C^infty-function, Probability Theory and Related Fields, June 1966, Volume 5, Issue 2, pp 173-174.
Wikipedia, Fabius function

Cf. A277429 (numerators), A272755, A272757, A272343.

c[0] = 1;
c[k_] := c[k] = Sum[((-1)^(k - r) c[r])/(1 + 2 k - 2 r)!, {r, 0, k - 1}]/(4^k - 1);
t[n_] := Mod[2 n + Sum[(-1)^Binomial[n, k], {k, 1, n}], 3];
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}]];
Table[Denominator[f[3/2^n]], {n, 2, 20}]

A288161 Denominator of half moments of Rvachëv function.

Original entry on oeis.org

2, 18, 6, 1350, 270, 23814, 17010, 65063250, 7229250, 9762090030, 4437313650, 8267713725521250, 635977978886250, 81188783595533250, 297692206516955250, 22510683177794610356250, 1564913803803903393750, 40011216302189267004656036250, 10529267447944543948593693750

Offset: 1

Juan Arias-de-Reyna, Jun 06 2017

a(n) is equal to the denominator of the integral over (0,1) of n*t^(n-1)*up(t).
These numbers are the half moments of the Rvachëv function. The Rvachëv function is related to the Fabius function, up(x)=F(x+1) for |x|<1 and  up(x)=0 for |x|>=1.
The sequence of numerators is not in the OEIS because it appears t coincide with A272755: Numerators of Fabius function F(1/2^n). In fact d(n) = n! 2^binomial(n,2)F(1/2^n). The coincidence depends on the fact that n! 2^binomial(n,2) divides the denominator of F(1/2^n). It is true that 2^binomial(n,2) divides this denominator, but I do not see any reason for n! to divide this denominator.

			The rationals d(n) are  1/2, 5/18, 1/6, 143/1350, 19/270,  ...

J. Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 [math.CA], 2017.
J. Arias de Reyna, Arithmetic of the Fabius function, arXiv:1702.06487 [math.NT], 2017.

Cf. A287936, A287937, A287938.

d[0] = 1;
d[n_] := d[n] =
  Sum[Binomial[n + 1, k] d[k], {k, 0, n - 1}]/((n + 1)*(2^n - 1));
Table[Denominator[d[n]], {n, 1, 20}]

Recurrence d(0)=1; d(n)=Sum_{k=0..n-1}(binomial(n+1,k)d(k))/((n+1)*(2^n-1)) with a(n) are the denominators of d(n).

It may also be defined to be the only sequence d(n) with d(0)=1 and such that the function f(x)=Sum_{n>=0} d(n) x^n/n! satisfies x*f(2x)=(e^x-1)*f(x).

A277471 Normalized values of the Fabius function: 2^binomial(n-1, 2) * (2n)! A005329(n) * F(1/2^n).

Original entry on oeis.org

2, 1, 5, 105, 7007, 1298745, 619247475, 723733375365, 2006532782969715, 12889909959143502285, 188494585656727188486375, 6188497678605937441851529425, 451101946262511157576785806552415, 72341127537387548941434093006996374625, 25326487488712595887856341442148826764706875

Offset: 0

Vladimir Reshetnikov, Oct 16 2016

nonn

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.
Juan Arias de Reyna proved that all terms are indeed integers. - Vladimir Reshetnikov, Feb 28 2017

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.

Juan Arias de Reyna, An infinitely differentiable function with compact support: Definition and properties, arXiv:1702.05442 [math.CA], 2017.
Juan Arias de Reyna, On the arithmetic of Fabius function, arXiv:1702.06487 [math.NT], 2017.
Yuri Dimitrov, G. A. Edgar, Solutions of Self-differential Functional Equations
G. A. Edgar, Examples of self differential functions
J. Fabius, A probabilistic example of a nowhere analytic C^infty-function, Probability Theory and Related Fields, June 1966, Volume 5, Issue 2, pp. 173-174.
Jan Kristian Haugland, Evaluating the Fabius function, arXiv:1609.07999 [math.GM], 23 Sep 2016.
Wikipedia, Fabius function

Cf. A005329, A272755, A272757.

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}]

a(n) = 2^binomial(n-1, 2) * (2*n)! * A005329(n) * A272755(n) / A272757(n).

cp's OEIS Frontend

A272757 Denominators of the Fabius function F(1/2^n).

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A287938 Integers associated with moments of Rvachëv function.

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

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A277430 Denominators of the Fabius function F(3/2^n).

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A288161 Denominator of half moments of Rvachëv function.

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A277471 Normalized values of the Fabius function: 2^binomial(n-1, 2) * (2*n)! * A005329(n) * F(1/2^n).

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A277471 Normalized values of the Fabius function: 2^binomial(n-1, 2) * (2n)! A005329(n) * F(1/2^n).