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A136559 G.f.: A(x) = Sum_{n>=0} arctanh( 2^(2n+1)*x )^(2n+1) / (2n+1)!; a power series in x with integer coefficients.

%I A136559 #15 Mar 16 2021 04:59:25
%S A136559 2,0,88,0,285088,0,112173964160,0,6667221644498203136,0,
%T A136559 66605167708510907980664608768,0,
%U A136559 120169056821375322042225614651624227643392,0,41233460218449924405779202537032142206549563511026450432,0,2796406262888046560966728498782777223041570797904775508376399120263413760
%N A136559 G.f.: A(x) = Sum_{n>=0} arctanh( 2^(2n+1)*x )^(2n+1) / (2n+1)!; a power series in x with integer coefficients.
%C A136559 2^n divides a(n) for n >= 0.
%H A136559 G. C. Greubel, <a href="/A136559/b136559.txt">Table of n, a(n) for n = 1..49</a>
%e A136559 G.f.: A(x) = 2*x + 88*x^3 + 285088*x^5 + 112173964160*x^7 + ...
%t A136559 Rest@With[{m=30}, CoefficientList[Series[Sum[ArcTanh[2^(2*j+1)*x]^(2*j+1)/(2*j + 1)!, {j,0,m+2}], {x,0,m}], x]] (* _G. C. Greubel_, Mar 15 2021 *)
%o A136559 (PARI) {a(n)=polcoeff(sum(k=0,n\2,atanh(2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!),n)}
%o A136559 (Magma)
%o A136559 m:=30;
%o A136559 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A136559 Coefficients(R!( (&+[Argtanh(2^(2*j+1)*x)^(2*j+1)/Factorial(2*j+1): j in [0..m+2]]) )); // _G. C. Greubel_, Mar 15 2021
%Y A136559 Cf. A136558.
%K A136559 nonn
%O A136559 1,1
%A A136559 _Paul D. Hanna_, Jan 10 2008