A214759 E.g.f. satisfies: A(x) = x - log(1-A(x))*log(1+A(x)).

1, 2, 12, 130, 1980, 38828, 931308, 26409696, 864340080, 32064955512, 1329619343712, 60942831972240, 3059502805907184, 166958986644198240, 9840268295795334528, 622945100439880885920, 42156597800781887551296, 3036974549700781049394240, 232047627350715685524211392

Offset

1,2

Links

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

Formula

E.g.f.: Series_Reversion(x + log(1-x)*log(1+x)).

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x))^n*log(1+x)^n / n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x))^n*log(1+x)^n/x / n! ).

a(n) ~ n^(n-1) * sqrt(r*(1-s^2) / (2+s + log(1-s)/(1+s) + log(1+s)/(1-s))) / (exp(n) * r^n), where s = 0.4236731503498759... is the root of the equation log(1+s)/(1-s) - log(1-s)/(1+s) = 1, and r = s + log(1-s)*log(1+s) = 0.2290093920590019... - Vaclav Kotesovec, Jan 13 2014

Example

E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 130*x^4/4! + 1980*x^5/5! +...
where A(x - log(1-x)*log(1+x)) = x and A(x) = x - log(1-A(x))*log(1+A(x)).
Related expansions:
-log(1-x)*log(1+x) = 2*x^2/2! + 10*x^4/4! + 188*x^6/6! + 7656*x^8/8! + 541152*x^10/10! + 58800960*x^12/12! +...
-log(1-A(x)) = x + 3*x^2/2! + 20*x^3/3! + 220*x^4/4! + 3374*x^5/5! + 66428*x^6/6! +...
log(1+A(x)) = x + x^2/2! + 8*x^3/3! + 88*x^4/4! + 1354*x^5/5! + 26708*x^6/6! +...
Other series:
A(x) = x - log(1-x)*log(1+x) + d/dx log(1-x)^2*log(1+x)^2/2! - d^2/dx^2 log(1-x)^3*log(1+x)^3/3! + d^3/dx^3 log(1-x)^4*log(1+x)^4/4! +...
log(A(x)/x) = -log(1-x)*log(1+x)/x + d/dx log(1-x)^2*log(1+x)^2/x/2! - d^2/dx^2 log(1-x)^3*log(1+x)^3/x/3! + d^3/dx^3 log(1-x)^4*log(1+x)^4/x/4! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x + Log[1-x]*Log[1+x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(x+log(1-x+x*O(x^n))*log(1+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, (-log(1-x+x*O(x^n)))^m*log(1+x+x*O(x^n))^m/m!)); n!*polcoeff(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, (-log(1-x+x*O(x^n)))^m*log(1+x+x*O(x^n))^m/x/m!))); n!*polcoeff(A, n)}

Crossrefs

Cf. A143154.

Sequence in context: A354493 A079199 A303926 * A365601 A185751 A090361

Adjacent sequences: A214756 A214757 A214758 * A214760 A214761 A214762

Keyword

nonn

Author

Paul D. Hanna, Jul 31 2012

Status

approved