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%I #14 Sep 12 2015 11:00:30
%S 1,3,20,208,2944,52748,1143876,29120292,851236008,28097170632,
%T 1033540588992,41922955409520,1858913275469808,89451843240275280,
%U 4642580506992107328,258501607230300771072,15370815613407241773312,972082967929423498417344,65152216036788689542930752,4613084984991245705602298688
%N E.g.f. satisfies: A'(x) = exp(A(x)) / (1 - A(x))^2.
%C Compare to: W'(x) = exp(W(x)) / (1 - W(x)), which holds when W(x) = -LambertW(-x).
%H Vaclav Kotesovec, <a href="/A257476/b257476.txt">Table of n, a(n) for n = 1..330</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of implicit functions if Fww = 0</a>
%F E.g.f.: Series_Reversion( Sum_{n>=1} -(n^2-n+1) * (-x)^n/n! ).
%F a(n) ~ GAMMA(1/3) * n^(n-5/6) / (3^(1/6) * sqrt(2*Pi) * (exp(1)-2)^(n-1/3)). - _Vaclav Kotesovec_, Apr 26 2015
%e E.g.f.: A(x) = x + 3*x^2/2! + 20*x^3/3! + 208*x^4/4! + 2944*x^5/5! +...
%e where
%e exp(A(x))/(1-A(x))^2 = 1 + 3*x + 20*x^2/2! + 208*x^3/3! + 2944*x^4/4! +...
%e The e.g.f. also satisfies:
%e x = Sum_{n>=1} (-1)^(n-1) * (n^2-n+1) * A(x)^n/n! , or
%e x = 1 - exp(-A(x)) * (1 + A(x)^2) [after Vaclav Kotesovec];
%e explicitly,
%e x = A(x)*x - 3*A(x)^2*x^2/2! + 7*A(x)^3*x^3/3! - 13*A(x)^4*x^4/4! + 21*A(x)^5*x^5/5! - 31*A(x)^6*x^6/6! +...
%t Rest[CoefficientList[InverseSeries[Series[1-E^(-x)*(1+x^2),{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Apr 26 2015 *)
%o (PARI) {a(n)=local(A=x); for(i=1,n, A = intformal((1-A)^-2*(exp(A+x*O(x^n)))) );n!*polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%o (PARI) {a(n)=local(A=x); A = serreverse(sum(m=1,n,(-1)^(m-1)*(m^2-m+1)*x^m/m!) +x*O(x^n)); n!*polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%K nonn
%O 1,2
%A _Paul D. Hanna_, Apr 25 2015
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