OFFSET
0,4
COMMENTS
Compare to: G(x) = exp( Integral (1 + 2*x*G(x) + x^2*G(x)^2)/G(x) dx ) holds when G(x) = 1/(1-x).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
E.g.f. satisfies: A'(x) = (1 - x^3*A(x)^3) / (1 - x*A(x)).
a(n) ~ n! * d^(n+3), where d = 0.9271503577507272... - Vaclav Kotesovec, Feb 24 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 15*x^4/4! + 64*x^5/5! + 355*x^6/6! +...
Related expansions:
A'(x) = 1 + x*A(x) + x^2*A(x)^2 = 1 + x + 4*x^2/2! + 15*x^3/3! + 64*x^4/4! + 355*x^5/5! + 2424*x^6/6! + 17521*x^7/7! +...
(1 + x*A(x) + x^2*A(x)^2)/A(x) = 1 + 3*x^2/2! + 2*x^3/3! + 23*x^4/4! + 36*x^5/5! + 673*x^6/6! + 1328*x^7/7! +...
log(A(x)) = x + 3*x^3/3! + 2*x^4/4! + 23*x^5/5! + 36*x^6/6! + 673*x^7/7! +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(intformal((1+x*A+x^2*A^2)/A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 14 2013
STATUS
approved