A287938 Integers associated with moments of Rvachëv function.

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

Offset: 0

Author

Juan Arias-de-Reyna, Jun 03 2017

Keywords

• nonn

Comments

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.

Links

• 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.

Crossrefs

Cf. A272755, A272757, A287936, A287937.

Programs

Mathematica

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 *)

Formula

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