A143922 - OEIS

%I #12 Feb 20 2014 12:44:06

%S 1,1,2,9,52,395,3666,40257,510600,7343523,118093310,2099660497,

%T 40896662124,866008634907,19808285169834,486698217317985,

%U 12784410332144656,357512156423101427,10604399352362692182

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

%C Compare definition of e.g.f. A(x) to the trivial statement:

%C if F(x) = 1/(1-x) then F(x) = 1 + x*exp(Integral F(x) dx).

%C Here Integral F(x) dx does not include the constant of integration.

%H Vaclav Kotesovec, <a href="/A143922/b143922.txt">Table of n, a(n) for n = 0..400</a>

%F E.g.f. derivative: A'(x) = [1 + x*A(x)^2]*(A(x) - 1)/x.

%F a(n) ~ n^n / (exp(n) * r^(n+1/2)), where r = 0.58963282569434540653295100228290669896338789564481715119... - _Vaclav Kotesovec_, Feb 20 2014

%e E.g.f. A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 395*x^5/5! +...

%e A(x)^2 = 1 + 2*x + 6*x^2/2! + 30*x^3/3! + 200*x^4/4! + 1670*x^5/5! +...

%e Let L(x) = Integral A(x)^2 dx where A(x) = 1 + x*exp(L(x)), then

%e L(x) = x + 2*x^2/2! + 6*x^3/3! + 30*x^4/4! + 200*x^5/5! +...

%e exp(L(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 79*x^4/4! + 611*x^5/5! +...

%t a = ConstantArray[0,20]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = (-n! * Sum[a[[i+1]] * a[[n-i]]/i!/(n-i-1)!,{i,0,n-1}] + n! * Sum[a[[k+1]]/k! * Sum[a[[i+1]]*a[[n-k-i]]/i!/(n-k-i-1)!,{i,0,n-1}],{k,0,n-1}])/(n-1),{n,2,19}]; a  (* _Vaclav Kotesovec_, Feb 20 2014 *)

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*exp(intformal(A^2)));n!*polcoeff(A,n)}

%Y Cf. A143923, A143924.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 06 2008