A227461 E.g.f. equals the series reversion of x - x*arctanh(x).

1, 2, 12, 128, 1920, 37104, 877184, 24520320, 791112960, 28932902400, 1182789053952, 53447706998784, 2645389044480000, 142326283714836480, 8270318699325112320, 516187815998727389184, 34440412737701955502080, 2446191865021002009477120, 184278436717136012676956160

Offset

1,2

Comments

Note that arctanh(x) = log((1+x)/(1-x))/2.

Links

Table of n, a(n) for n=1..19.

Formula

E.g.f. A(x) satisfies:

(1) A(x - x*arctanh(x)) = x.

(2) A(x) = x/(1 - arctanh(A(x))).

(3) A(x) = tanh( (A(x)-x)/A(x) ).

(4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * arctanh(x)^n / n!.

(5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * arctanh(x)^n / n! ).

a(n) ~ n^(n-1) * s^2 * (1/s^2-1)^(n+1/2) / (exp(n) * sqrt(2)), where s = 0.43415423687337693781... is the root of the equation (1-s^2)*(1-arctanh(s)) = s. - Vaclav Kotesovec, Jan 13 2014

Example

E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ...
where A(x) = x/(1 - arctanh(A(x))).
The e.g.f. satisfies:
(4) A(x) = x + x*arctanh(x) + d/dx x^2*arctanh(x)^2/2! + d^2/dx^2 x^3*arctanh(x)^3/3! + d^3/dx^3 x^4*arctanh(x)^4/4! + ...
(5) log(A(x)/x) = arctanh(x) + d/dx x*arctanh(x)^2/2! + d^2/dx^2 x^2*arctanh(x)^3/3! + d^3/dx^3 x^3*arctanh(x)^4/4! + ...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - x*ArcTanh[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(x-x*atanh(x +x*O(x^n))), n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*atanh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*atanh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A227460.

Sequence in context: A143136 A214224 A214431 * A367374 A367853 A228608

Adjacent sequences: A227458 A227459 A227460 * A227462 A227463 A227464

Keyword

nonn

Author

Paul D. Hanna, Jul 13 2013

Status

approved