A143923 - OEIS

%I #6 Feb 28 2014 03:38:38

%S 1,1,2,12,88,860,10392,149044,2478752,46875492,993291880,23311581524,

%T 600207989808,16820818373476,509711184710840,16606143020005620,

%U 578830045479469120,21493718211307208420,847057099952645864712

%N E.g.f. A(x) satisfies: A(x) = 1 + x*exp(Integral A(x)^3 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.

%C Limit n->infinity (a(n)/n!)^(1/n) = 2.274991... - _Vaclav Kotesovec_, Feb 28 2014

%H Vaclav Kotesovec, <a href="/A143923/b143923.txt">Table of n, a(n) for n = 0..176</a>

%F E.g.f. derivative: A'(x) = [1 + x*A(x)^3]*(A(x) - 1)/x.

%e E.g.f. A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 88*x^4/4! + 860*x^5/5! +...

%e A(x)^3 = 1 + 3*x + 12*x^2/2! + 78*x^3/3! + 696*x^4/4! + 7740*x^5/5! +...

%e Let L(x) = Integral A(x)^3 dx where A(x) = 1 + x*exp(L(x)), then

%e L(x) = x + 3*x^2/2! + 12*x^3/3! + 78*x^4/4! + 696*x^5/5! +...

%e exp(L(x)) = 1 + x + 4*x^2/2! + 22*x^3/3! + 172*x^4/4! + 1732*x^5/5! +...

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*exp(intformal(A^3)));n!*polcoeff(A,n)}

%Y Cf. A143922, A143924.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 06 2008