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A225052 E.g.f. satisfies: A(x) = exp( Integral 1/(1 - x*A(x)) dx ).

%I A225052 #19 Feb 16 2025 08:33:19
%S A225052 1,1,2,8,50,426,4606,60418,932282,16547562,332152614,7439791314,
%T A225052 183964790514,4977606096570,146287199495310,4640510332052370,
%U A225052 158035939351814250,5750979655319685834,222710142933114209526,9144799526131421284434,396863889188887568805282
%N A225052 E.g.f. satisfies: A(x) = exp( Integral 1/(1 - x*A(x)) dx ).
%C A225052 Compare to: W(x) = exp( Integral W(x)/(1 - x*W(x)) dx ), which is satisfied by:  W(x) = LambertW(-x)/(-x)  =  Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
%C A225052 Compare to: C(x) = exp( Integral C(x)^2/(1 - x*C(x)^2) dx ), which is satisfied by: C(x) = (1-sqrt(1-4*x))/(2*x) (Catalan numbers, A000108).
%H A225052 Vaclav Kotesovec, <a href="/A225052/b225052.txt">Table of n, a(n) for n = 0..390</a>
%H A225052 Eric Weisstein, <a href="https://mathworld.wolfram.com/ExponentialIntegral.html">MathWorld: Exponential Integral</a>
%F A225052 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!  satisfies:
%F A225052 (1) 1/(1 - x*A(x)) = 1 + Sum_{n>=1} n*a(n)*x^n/n!.
%F A225052 (2) log(A(x)) = x + Sum_{n>=1} n*a(n)*x^(n+1)/(n+1)!.
%F A225052 (3) log(A(x)) = Integral Sum_{n>=1} n!*(x*A(x))^(n-1) * Product_{k=1..n} 1/(1 + k*x*A(x)) dx. - _Paul D. Hanna_, Jun 07 2014
%F A225052 E.g.f. derivative: A'(x) = A(x) / (1-x*A(x)). - _Vaclav Kotesovec_, Feb 19 2014
%F A225052 a(n) ~ n^(n-1) / (exp(n) * r^(n+1/2)), where r = 0.4271853687986028467... is the root of the equation Ei(1/r) - Ei(1) = r*exp(1/r), where Ei is the Exponential integral. - _Vaclav Kotesovec_, Feb 19 2014
%e A225052 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 50*x^4/4! + 426*x^5/5! +...
%e A225052 where
%e A225052 (1) 1/(1 - x*A(x)) = 1 + x + 4*x^2/2! + 24*x^3/3! + 200*x^4/4! + 2130*x^5/5! + 27636*x^6/6! +...+ n*a(n)*x^n/n! +...
%e A225052 (2) log(A(x)) = x + x^2/2! + 4*x^3/3! + 24*x^4/4! + 200*x^5/5! + 2130*x^6/6! + 27636*x^7/7! +...+ n*a(n)*x^(n+1)/(n+1)! +...
%e A225052 (3) A'(x)/A(x) = 1/(1+x*A(x)) + 2!*x*A(x)/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^2*A(x)^2/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) +... = 1/(1-x*A(x)).
%t A225052 a = ConstantArray[0,20]; a[[1]]=1; Do[a[[n+1]] = a[[n]] + n!*(a[[n]]/(n-1)! + Sum[a[[i]]*a[[n-i]]/i!/(n-i-1)!,{i,1,n-1}]),{n,1,19}]; Flatten[{1,a}] (* _Vaclav Kotesovec_, Feb 19 2014 *)
%t A225052 FindRoot[ExpIntegralEi[1/r] - ExpIntegralEi[1] == r*E^(1/r),{r,1/2},WorkingPrecision->50] (* program for numerical value of the radius of convergence r, _Vaclav Kotesovec_, Feb 19 2014 *)
%o A225052 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(intformal(1/(1-x*A +x*O(x^n)))));n!*polcoeff(A,n)}
%o A225052 for(n=0,20,print1(a(n),", "))
%Y A225052 Cf. A091725.
%K A225052 nonn
%O A225052 0,3
%A A225052 _Paul D. Hanna_, Apr 26 2013