[seqfan] Fw: Closed form?

[David Wilson]

The reason I ask.

I was once asked about the asymptotic growth of A000123.

A recurrence for A000123 is a(n) = a(n-1) + a([n/2]).  This gives 
a(n)-a(n-1) = a([n/2]). One might hope to approximate a with real function f 
satisfying f'(n) = f(n/2). Setting f(0) = a(0) = 1, power series analysis of 
f'(n) = f(n/2) gives the power series below.

f is encouragingly close to A000123 for small values of f (<= 10000) but 
drops away for larger values. Maybe someone with better math software can 
fudge the equations. Maybe f'(n-1/2) = f(n/2) is closer.

----- Original Message ----- 
From: "David Wilson" <dwilson at gambitcomm.com>
To: "Sequence Fanatics" <seqfan at list.seqfan.eu>
Sent: Wednesday, April 29, 2009 11:26 AM
Subject: [seqfan] Closed form?


> Is there a closed form for
>
> f(x) = SUM(k = 0 to inf; x^k/(2^((k^2-k)/2)*k!)))
>
> i.e;
>
> f(x) = 1 + x + x^2/4 + x^3/48 + x^4/1536 + x^5/122880 + ...
>
>
>
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/


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