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A194563
G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A_{n-1}(x)^n where A_{n}(x) = A_{n-1}(x) + x^n*A_{n-1}(x)^n for n>0 with A_0(x)=1.
1
1, 1, 1, 3, 5, 11, 29, 71, 172, 429, 1109, 2920, 7730, 20589, 55059, 147881, 399684, 1086523, 2968414, 8145414, 22432134, 61964161, 171621042, 476482795, 1325825751, 3696818803, 10328130168, 28908075633, 81053999054, 227636478284, 640289071629, 1803570718863, 5087114281297
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
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
Sequence in context: A326392 A213210 A279674 * A080443 A168607 A057735
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 28 2011
STATUS
approved