OFFSET
0,4
COMMENTS
What is the limit a(n+1)/a(n)?
Compare to the g.f. of the Catalan numbers: C(x) = Sum_{n>=0} x^n*C(x)^n.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1001
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 29*x^6 + 71*x^7 +...
The g.f. equals the series
A(x) = 1 + x*A_0(x) + x^2*A_1(x)^2 + x^3*A_2(x)^3 + x^4*A_3(x)^4 +...
where
A_{n}(x) = A_{n-1}(x) + x^n*A_{n-1}(x)^n,
as illustrated by:
A_0 = 1,
A_1 = 1 + x,
A_2 = 1 + x*(1) + x^2*(1 + x)^2,
A_3 = 1 + x*A_0 + x^2*A_1^2 + x^3*A_2^3,
A_4 = 1 + x*A_0 + x^2*A_1^2 + x^3*A_2^3 + x^4*A_3^4,
A_5 = 1 + x*A_0 + x^2*A_1^2 + x^3*A_2^3 + x^4*A_3^4 + x^5*A_4^5, ...
More explicitly,
A(x) = 1 + x*(1) + x^2*(1 + x)^2 + x^3*(1 + x + x^2*(1 + x)^2)^3 + x^4*(1 + x + x^2*(1 + x)^2 + x^3*(1 + x + x^2*(1 + x)^2)^3)^4 +...
The value a(n+1)/a(n) tends to a limit:
a(101)/a(100) = 2.89958617222...
a(201)/a(200) = 2.91785966732...
a(301)/a(300) = 2.92396128279...
a(401)/a(400) = 2.92700772800...
a(501)/a(500) = 2.92883239050...
a(901)/a(900) = 2.93206543661...
a(1001)/a(1000) = 2.93246807951...
PROG
(PARI) {a(n) = local(A=1); for(m=1, n, A = A + (x*A + x*O(x^n))^m ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 28 2011
STATUS
approved