A273400 a(n) = Catalan(Catalan(Catalan(n))).

1, 1, 2, 39044429911904443959240

Offset: 0

Author

Waldemar Puszkarz, May 21 2016

Keywords

• nonn

Comments

a(4) has 1610164 digits and it is thus too large to be included.

Conjecture. The number of digits of a(n) grows asymptotically faster than Catalan(n), i.e., only a finite number of terms of a(n) has the number of digits less than the value of Catalan(n).

This also appears to be true for the Fibonacci sequence (A000045) and the sequence of powers of 2 (A000079): it takes two additional iterations of these sequences for the number of digits of these iterated sequences to grow faster than the corresponding original sequences. However, it appears that it takes only one additional iteration of the factorial (A000142) for this to happen.

The number of digits of a(n) grows asymptotically faster than Fibonacci(n), but that is already true for Catalan(Catalan(n)) (A273399).

Examples

			For n = 2, a(2) = Catalan(Catalan(Catalan(2))) = Catalan(Catalan(2)) = Catalan(2) = 2 as Catalan(2) = 2.
		

Crossrefs

Cf. A000108 (Catalan), A273399 (Catalan(Catalan(n))), A058051.

Programs

Maple

a:= ((n-> binomial(2*n, n)/(n+1))@@3):
seq(a(n), n=0..3);  # Alois P. Heinz, May 27 2025

Mathematica

CatalanNumber[CatalanNumber[CatalanNumber[Range[0, 3]]]]

PARI

for(n=0, 3, cn=binomial(2*n, n)/(n+1); cn2=binomial(2*cn, cn)/(cn+1); cn3=binomial(2*cn2, cn2)/(cn2+1); print1(cn3 ", "))

Formula

a(n) = A000108(A000108(A000108(n))).