[seqfan] Re: Fw: Closed form?

Maximilian Hasler maximilian.hasler at gmail.com
Thu Apr 30 21:53:14 CEST 2009

On Thu, Apr 30, 2009 at 2:59 PM, Prof. Dr. Alois Heinz
<heinz at hs-heilbronn.de> wrote:
> I have looked at a larger range now, and a(n)/f(n) is drifting slowly
> upwards or downwards, always with prefix .9233

Don't you think that this "drifting up and down" corresponds to the
periodic variation in the Phi function as per Philippe's comment
[quoted below], and thus won't disappear ?

> Digits:=160:

^^^ I don't understand this, if we are only interested in the first 10
digits of the ratio a(n)/f(n).
Would Maple compute the first 10 digits of f(x) incorrectly without this ?


Comment from Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Sep 06
2008 (Start): The asymptotic rate of growth is known precisely - see
De Bruijn's paper. With p(n) the number of partitions of n into powers
of two, the asymptotic formula of de Bruijn is:

log(p(2*n)) = 1/(2*L2)*(log(n/log(n)))^2 + (1/2 + 1/L2 +
LL2/L2)*log(n) - (1 + LL2/L2)*log(log(n)) + Phi(log(n/log(n))/L2),

where L2=log(2), LL2=log(log(2)) and Phi(x) is a certain periodic
function with period 1 and a tiny amplitude.

Numerically, Phi(x) appears to have a mean value around 0.66. An
expansion up to O(1) term had been obtained earlier by Kurt Mahler.

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