Re: -- A sequence or real numbers

On 13 Feb, 19:30, c...@xxxxxxxxxxxxxxx wrote:
Does anyone know, or at least have a clue on the following question ?

Let (b_n)_n be the sequence defined by :

b_1=1/4 , b_2=1/16 and

b_{n+1}=b_1 b_n+b_2 b_{n-1}+ ... +b_k b_{n-k+1}+ ... +b_1 b_n

Ex: b_3=b_1 b_2+b_2 b_1=1/32
b_4=b_1 b_3+b_2 b_2+b_3 b_1=5/256 etc.


This is a variant of the well-known recurrence for Catalan numbers:
b_n = 4^{-n} C_{n-1}
where C_n = (2n)!/(n!(n+1)!).

Finally, let

X_n = b_n . n^{3/2}

From Stirling's formula, C_n ~ 4^n/(n^{3/2}pi^{1/2})
so X_n -> 4/pi^{1/2} as n --> infinity.

Victor Meldrew
"I don't believe it!"

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