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

1, 1, 2, 8, 50, 426, 4606, 60418, 932282, 16547562, 332152614, 7439791314, 183964790514, 4977606096570, 146287199495310, 4640510332052370, 158035939351814250, 5750979655319685834, 222710142933114209526, 9144799526131421284434, 396863889188887568805282

Offset: 0

Author

Paul D. Hanna, Apr 26 2013

Keywords

• nonn

Comments

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!.

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).

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 50*x^4/4! + 426*x^5/5! +...
where
(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! +...
(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)! +...
(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)).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..390
• Eric Weisstein, MathWorld: Exponential Integral

Crossrefs

Cf. A091725.

Programs

Mathematica

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 *)
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 *)

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)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:

(1) 1/(1 - x*A(x)) = 1 + Sum_{n>=1} n*a(n)*x^n/n!.

(2) log(A(x)) = x + Sum_{n>=1} n*a(n)*x^(n+1)/(n+1)!.

(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

E.g.f. derivative: A'(x) = A(x) / (1-x*A(x)). - Vaclav Kotesovec, Feb 19 2014

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