A136559 -id:A136559 - cp's OEIS frontend

A136558 G.f.: A(x) = Sum_{n>=0} arcsinh( 2^(2n+1)*x )^(2n+1) / (2n+1)!; a power series in x with integer coefficients.

2, 0, 84, 0, 276892, 0, 111457917800, 0, 6660816097416169260, 0, 66597307693046550483175282456, 0, 120167520447600665027319450022840022638104, 0, 41233407800231936275686869695450406221641586822849599440, 0, 2796405930832642696090353299413183601303402593622351242536692586333202380, 0

Offset: 1

Paul D. Hanna, Jan 10 2008

nonn

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

G. C. Greubel, Table of n, a(n) for n = 1..49

Cf. A136559.

m:=30;
R:=PowerSeriesRing(Rationals(), 30);
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

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 *)

{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)}

A136749 G.f.: Sum_{n>=0} arctanh(2^n*x)^n / n!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 8, 88, 2816, 285088, 96376832, 112173964160, 458290670993408, 6667221644498203136, 349410482551421802119168, 66605167708510907980664608768, 46557944823739673536754738305957888, 120169056821375322042225614651624227643392

Offset: 0

Paul D. Hanna, Jan 21 2008

nonn

This is a special application of the following identity.
Let F(x),G(x), be power series in x such that F(0)=1,G(0)=1, then
Sum_{n>=0} m^n * H(q^n*x) * log( F(q^n*x)*G(x) )^n / n! =
Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n] H(y)*F(y)^(m*q^n).

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 88*x^3 + 2816*x^4 + 285088*x^5 + 96376832*x^6 + ...
where
A(x) = 1 + arctanh(2*x) + arctanh(2^2*x)^2/2! + arctanh(2^3*x)^3/3! + arctanh(2^4*x)^4/4! + ...

Cf. A136559, A136647.

{a(n)=polcoeff(sqrt((1+x)/(1-x +x*O(x^n)))^(2^n),n)}

{a(n)=polcoeff(exp(2^n*atanh(x +x*O(x^n))), n)}

{a(n)=polcoeff(sum(k=0,n,atanh(2^k*x +x*O(x^n))^k/k!),n)}

{a(n)=polcoeff(sum(k=0,n,log((1+2^k*x)/(1-2^k*x +x*O(x^n)))^k/(2^k*k!)),n)}

a(n) = [y^n] sqrt((1+y)/(1-y))^(2^n) for n >= 0.
a(n) = n!*[x^n] exp( 2^n*arctanh(x) ).
G.f.: Sum_{n>=0} log( (1 + 2^n*x)/(1 - 2^n*x) )^n /(2^n*n!).

cp's OEIS Frontend

A136558 G.f.: A(x) = Sum_{n>=0} arcsinh( 2^(2n+1)*x )^(2n+1) / (2n+1)!; a power series in x with integer coefficients.

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A136749 G.f.: Sum_{n>=0} arctanh(2^n*x)^n / n!, a power series in x with integer coefficients.

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