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A287938 Integers associated with moments of Rvachëv function.

%I A287938 #32 May 22 2025 09:54:23
%S A287938 1,1,19,2915,2788989,14754820185,402830065455939,54259734183964303995,
%T A287938 34931036957548128175343565,104968042559556881090071537121985,
%U A287938 1445701512369903326110289606343988638195,89942525814858602265845303890518923811304544595,24979493321562411847493262443987087581059026281953954525
%N A287938 Integers associated with moments of Rvachëv function.
%C A287938 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.
%H A287938 J. 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 A287938 J. Arias de Reyna, <a href="https://arxiv.org/abs/1702.06487">Arithmetic of the Fabius function</a>, arXiv:1702.06487 [math.NT], 2017.
%F A287938 a(n) = (2*n+1)!!*(Product_{k=1..n} (2^(2*k)-1))*A287936(n)/A287937(n).
%t A287938 c[0] = 1;
%t A287938 c[n_] := c[n] =
%t A287938    Sum[Binomial[2 n + 1, 2 k] c[k], {k, 0, n - 1}]/((2 n + 1) (2^(2 n) - 1));
%t A287938 a[n_] := a[n] = c[n] (2 n + 1)!! Product[(2^(2 k) - 1), {k, 1, n}];
%t A287938 Table[a[n], {n, 0, 30}]
%t A287938 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 *)
%Y A287938 Cf. A272755, A272757, A287936, A287937.
%K A287938 nonn
%O A287938 0,3
%A A287938 _Juan Arias-de-Reyna_, Jun 03 2017