A136647 -id:A136647 - cp's OEIS frontend

A141367 G.f.: Sum_{n>=0} arcsinh(4^n*x)^n/n!, a power series in x having only integer coefficients.

1, 4, 128, 43680, 178946048, 9382409745280, 6558834518571089920, 62879485860387254833099776, 8439542720341303996200869198561280, 16110026846830031883594370688522189192189952

Offset: 0

Paul D. Hanna, Jul 02 2008

nonn

			G.f.: A(x) = 1 + 4*x + 128*x^2 + 43680*x^3 + 178946048*x^4 + ...

G. C. Greubel, Table of n, a(n) for n = 0..40

Cf. A136647, A141368.

Table[SeriesCoefficient[(Sqrt[1 + x^2] + x)^(4^n), {x, 0, n}], {n, 0, 25}] (* G. C. Greubel, Apr 15 2017 *)

{a(n)=polcoeff(sum(k=0,n, asinh(4^k*x +x*O(x^n))^k/k!),n)}

{a(n)=polcoeff((x+sqrt(1+x^2 +x*O(x^n)))^(4^n),n)}

a(n) = [x^n] [ sqrt(1+x^2) + x ]^(4^n).
More generally, the following coefficient of x^n in the series:
[x^n] Sum_{n>=0} arcsinh(q^n*x)^n/n! = [x^n] [ sqrt(1+x^2) + x ]^(q^n) is an integer for any even integer q.

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!).

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A141367 G.f.: Sum_{n>=0} arcsinh(4^n*x)^n/n!, a power series in x having only 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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