|
%I #7 Jan 13 2014 09:13:59
%S 1,2,18,258,5165,132805,4171552,154816396,6628496163,321607254643,
%T 17438557525290,1045054518319394,68589806347167547,
%U 4893053653878222677,376976677153445288012,31194322614945877275400,2759269697125674797536075,259812433080660490261356447
%N E.g.f. satisfies: A(x) = x - (exp(A(x)) - 1) * log(1 - A(x)).
%F E.g.f. satisfies:
%F (1) A(x) = Series_Reversion(x + (exp(x)-1)*log(1-x) ).
%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n*(-log(1-x))^n / n!.
%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n*(-log(1-x))^n/x / n! ).
%F 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
%e E.g.f: A(x) = x + 2*x^2/2! + 18*x^3/3! + 258*x^4/4! + 5165*x^5/5! +...
%e where A(x + (exp(x)-1)*log(1-x)) = x.
%e The e.g.f. satisfies the series:
%e 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! +...
%e Logarithmic series:
%e 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! +...
%t Rest[CoefficientList[InverseSeries[Series[x + (Exp[x]-1)*Log[1-x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 13 2014 *)
%o (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)}
%o for(n=1, 21, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {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)}
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {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)}
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 08 2012
| 