A287937 -id:A287937 - cp's OEIS frontend

A287936 Numerator of moments of Rvachëv function up(x).

1, 1, 19, 583, 132809, 46840699, 4068990560161, 1204567303451311, 4146897304424408411, 18814360006695807527868793, 21431473463327429953796293981397, 911368783375270623395381542054690099, 3805483535214088799368825731508632105336401423

Offset: 0

Juan Arias-de-Reyna, Jun 03 2017

a(n)/A287937(n) is equal to the integral of t^(2n) * up(t), 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. A287937, A287938.

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));
Table[Numerator[c[n]], {n, 0, 30}]

Recurrence c(0)=1, c(n)=Sum_{k=0..n-1}(binomial(2n+1,2k) c_k)/((2n+1)*(2^(2n)-1)), where c(n)=a(n)/A287937(n).

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).

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).

A288163 Integers related to the half moments of Rvachëv function.

Original entry on oeis.org

1, 1, 5, 84, 4004, 494760, 150120600, 107969547840, 179605731622464, 678695382464158080, 5745964983105758544000, 107798142804281290451059200, 4441362930723337358985334172160, 398854836980938754158182857661404160, 77576833096847783279235708819073596288000

Offset: 0

Juan Arias-de-Reyna, Jun 06 2017

nonn

These numbers determine 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.

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, A288161.

d[0] = 1;
d[n_] := d[n] =
  Sum[Binomial[n + 1, k] d[k], {k, 0, n - 1}]/((n + 1)*(2^n - 1));
a[n_] := (n + 1)! Product[(2^k - 1), {k, 1, n}] d[n];
Table[a[n], {n, 0, 14}]

a(n) = (n+1)!*Product_{k=1..n}(2^k-1)*d(n) where d(n) are the rationals defined by the recurrence d(0)=1; d(n)=Sum_{k=0..n-1}[binomial(n+1,k)d(k)]/((n+1)*(2^n-1)) (cf. A288161).

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A287936 Numerator of moments of Rvachëv function up(x).

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A287938 Integers associated with moments of Rvachëv function.

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A288161 Denominator of half moments of Rvachëv function.

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A288163 Integers related to the half moments of Rvachëv function.

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