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

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

Offset

0,2

Comments

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).

Links

Table of n, a(n) for n=0..13.

Formula

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!).

Example

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! + ...

Prog

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

Crossrefs

Cf. A136559, A136647.

Sequence in context: A141313 A009144 A132316 * A226321 A054955 A012299

Adjacent sequences: A136746 A136747 A136748 * A136750 A136751 A136752

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2008

Status

approved