OFFSET
0,3
COMMENTS
Compare the definition of the e.g.f. A(x) to the trivial statement:
if F(x) = 1/(1-x) then F'(x) = (1 + x*F(x))^2.
In general, if e.g.f satisfies A'(x) = (1+x*A(x))^p, then a(n) ~ c(p) * d(p)^n * n! / n^(1-1/(p-1)), where c(p) and d(p) are constants independent on n. - Vaclav Kotesovec, Jul 15 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..250
FORMULA
E.g.f. satisfies: A(x) = 1 + Integral (1 + x*A(x))^4 dx.
a(n) ~ c * n^(n-1/6) / (exp(n) * r^n), where r = 0.475460695778... and c = 2.2399022393... . - Vaclav Kotesovec, Jul 14 2014
MATHEMATICA
n = 18; A = 1+x; Do[A = 1 + Integrate[(1+x*A)^4 + O[x]^n, x], {i, 0, n}]; CoefficientList[A, x]*Range[0, n]! (* Jean-François Alcover, Jul 20 2017, adapted from PARI *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal((1+x*A+x*O(x^n))^4)); n!*polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 10 2008
STATUS
approved