A224788 - OEIS

%I #22 Feb 20 2014 10:41:02

%S 1,1,3,18,168,2142,34704,682740,15810372,421339176,12702393792,

%T 427435993512,15881634963216,645804320863680,28527455317884336,

%U 1360332028008819360,69645942884911181184,3810436222004101378656,221867131720533800409216,13698420738298341356760768

%N E.g.f. satisfies: A(x) = exp( Integral A(x)/(1 - x*A(x)^2) dx ).

%C 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).

%C 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!.

%H Vaclav Kotesovec, <a href="/A224788/b224788.txt">Table of n, a(n) for n = 0..300</a>

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">MathWorld: Exponential Integral</a>

%F E.g.f. derivative: A'(x) = A(x)^2 / (1-x*A(x)^2). - _Vaclav Kotesovec_, Feb 19 2014

%F a(n) ~ n^(n-1) / (sqrt(2) * exp(n) * r^(n+1/4)), where r = 0.28396034297... is the root of the equation Ei(1/sqrt(r)) - Ei(1) + exp(1) = (r+sqrt(r)) * exp(1/sqrt(r)), where Ei is the Exponential Integral. - _Vaclav Kotesovec_, Feb 19 2014

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 18*x^3/3! + 168*x^4/4! + 2142*x^5/5! +...

%e where

%e log(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 99*x^4/4! + 1236*x^5/5! + 19752*x^6/6! +...

%e A(x)/(1-x*A(x)^2) = 1 + 2*x + 11*x^2/2! + 99*x^3/3! + 1236*x^4/4! + 19752*x^5/5! +...

%t a = ConstantArray[0,21]; a[[1]]=1; a[[2]]=1; Do[a[[n+2]] = n!*Sum[a[[i+1]]*a[[n-i+1]]/i!/(n-i)!,{i,0,n}] + n!*Sum[a[[j+1]]/(j-1)!*Sum[a[[i+1]]*a[[n-j-i+1]]/i!/(n-j-i)!,{i,0,n}],{j,1,n}],{n,1,18}]; a (* _Vaclav Kotesovec_, Feb 19 2014 *)

%t FindRoot[ExpIntegralEi[1/Sqrt[r]] - ExpIntegralEi[1] + E == (r+Sqrt[r]) * E^(1/Sqrt[r]),{r,1/2},WorkingPrecision->50] (* program for numerical value of the radius of convergence r, _Vaclav Kotesovec_, Feb 19 2014 *)

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(intformal(A/(1-x*A^2 +x*O(x^n)))));n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A225052, A091725.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 28 2013