Re: -- A sequence or real numbers

cosv@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.

Finally, let

X_n=b_n . n^{3/2}

A computer evaluation hints to the fact that (X_n) is decreasing to a
limit approx equal to 1.141...

Proof (?)

Note: the initial values (i.e. b_1,b_2) seems to be essential. Any
change, and (X_n)_n appears
to behave chaotically. [again, computer says so ]

Any information is warmly welcome.
Thanks
cosv


Your sequence seems to be close to the sequence of Catalan numbers
c_n=binomial(2n, n)/(n+1). Indeed I would say that your b_n is
c_{n-1}/4^n.
From Stirling formula n! = (n^n sqrt(2 Pi n)/e^n) (1 + o(1))
you should come
to a formula for your limit involving sqrt(Pi).

Sincerely

.

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