A215362 E.g.f. satisfies: A(x) = x - (exp(A(x)) - 1) * log(1 - A(x)).

1, 2, 18, 258, 5165, 132805, 4171552, 154816396, 6628496163, 321607254643, 17438557525290, 1045054518319394, 68589806347167547, 4893053653878222677, 376976677153445288012, 31194322614945877275400, 2759269697125674797536075, 259812433080660490261356447

Offset

1,2

Links

Table of n, a(n) for n=1..18.

Formula

E.g.f. satisfies:

(1) A(x) = Series_Reversion(x + (exp(x)-1)*log(1-x) ).

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

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

a(n) ~ n^(n-1) * (1-s)*sqrt(1/(exp(s)*(3-2*s) - 1 - exp(s)*(1-s)^2*log(1-s))) / (exp(n) * r^(n-1/2)), where s = 0.3036870503596169812... is the root of the equation (2-s-exp(s))/(1-s) + exp(s)*log(1-s) = 0, and r = s + (exp(s)-1)*log(1-s) = 0.1752487484407525433... - Vaclav Kotesovec, Jan 13 2014

Example

E.g.f: A(x) = x + 2*x^2/2! + 18*x^3/3! + 258*x^4/4! + 5165*x^5/5! +...
where A(x + (exp(x)-1)*log(1-x)) = x.
The e.g.f. satisfies the series:
A(x) = x - (exp(x)-1)*log(1-x) + d/dx (exp(x)-1)^2*log(1-x)^2/2! - d^2/dx^2 (exp(x)-1)^3*log(1-x)^3/3! + d^3/dx^3 (exp(x)-1)^4*log(1-x)^4/4! +...
Logarithmic series:
log(A(x)/x) = -(exp(x)-1)*log(1-x)/x + d/dx (exp(x)-1)^2*log(1-x)^2/x/2! - d^2/dx^2 (exp(x)-1)^3*log(1-x)^3/x/3! + d^3/dx^3 (exp(x)-1)^4*log(1-x)^4/x/4! +...

Mathematica

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

Prog

(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x + (exp(x+x*O(x^n))-1)*log(1-x+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=1, 21, 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)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*(-log(1-x+x*O(x^n)))^m)/m!)+x*O(x^n)); 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)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*(-log(1-x+x*O(x^n)))^m/x)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Crossrefs

Sequence in context: A213643 A143138 A151362 * A360974 A350461 A099880

Adjacent sequences: A215359 A215360 A215361 * A215363 A215364 A215365

Keyword

nonn

Author

Paul D. Hanna, Aug 08 2012

Status

approved