OFFSET
0,3
COMMENTS
Compare to the trivial identity for the Catalan function C(x) = 1 + x*C(x)^2:
C(x) = Sum_{n>=0} x^n*C(x)^n * Sum_{k=0..n} binomial(n,k)*x^k*(1-x)^(n-k).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..170
FORMULA
a(n) ~ c * d^n / n^(3/2), where d = 4.871479127250632..., c = 0.4392903421166... . - Vaclav Kotesovec, Jul 05 2014
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 24*x^4 + 86*x^5 + 330*x^6 + 1311*x^7 +...
where g.f. A(x) satisfies:
A(x) = 1 + x*A(x)*((1-x) + x)
+ x^2*A(x)^2*((1-x)^2 + 2^2*x*(1-x) + x^2)
+ x^3*A(x)^3*((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3)
+ x^4*A(x)^4*((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4)
+ x^5*A(x)^5*((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5) +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*A^m*sum(k=0, m, binomial(m, k)^2*x^k*(1-x)^(m-k)) +x*O(x^n)));; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 31 2013
STATUS
approved