A136558 - OEIS

%I #14 Mar 16 2021 04:59:07

%S 2,0,84,0,276892,0,111457917800,0,6660816097416169260,0,

%T 66597307693046550483175282456,0,

%U 120167520447600665027319450022840022638104,0,41233407800231936275686869695450406221641586822849599440,0,2796405930832642696090353299413183601303402593622351242536692586333202380,0

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

%H G. C. Greubel, <a href="/A136558/b136558.txt">Table of n, a(n) for n = 1..49</a>

%e G.f.: A(x) = 2*x + 84*x^3 + 276892*x^5 + 111457917800*x^7 + ...

%t Rest@With[{m=30}, CoefficientList[Series[Sum[ArcSinh[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 (PARI) {a(n)=polcoeff(sum(k=0,n\2, asinh(2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!),n)}

%o (Magma)

%o m:=30;

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o Coefficients(R!( (&+[Argsinh(2^(2*j+1)*x)^(2*j+1)/Factorial(2*j+1): j in [0..m+2]]) )); // _G. C. Greubel_, Mar 15 2021

%Y Cf. A136559.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jan 10 2008