Time for some exponentials
Jan. 8th, 2012 08:22 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoyEDIT: D'oh. Miswrote some sums so as to screw things up. Fixed now.
Let's consider the sequence an=sum from k=0 to n of 1/k! * (n-k)^k.
We can round this down to bn=sum from k=0 to n of (n-k choose k). If we do that, we get bn=Fn+1, where Fn is the n'th Fibonacci number. (This is a well-known sum.) Or, in other words, about φn+1/√5, where φ is the golden ratio.
On the other hand, we can round it up to cn=sum from k=0 to n of (n-k multichoose k). If we do that, we get cn=2n-1 for n≥1 (while c0=1). (This is easy.)
So the original sequence an, being somewhere between these two, is evidently about exponential, with base somewhere inbetween φ and 2. Indeed, running the numbers, it is apparently about
1/(1+W(1))*1/W(1)n, or, equivalently, 1/(1+W(1))*eW(1)n
where W is the Lambert W function (i.e. W(1) is the unique real number such that 1/W(1)=eW(1)). Our base here, 1/W(1) (or eW(1)), is about 1.76322283...
So, that's kind of neat, but I have no idea why this might be. I don't really feel like thinking about it now, though, so I thought I'd just throw it out there and see if anyone knew anything about it...
-Harry
Let's consider the sequence an=sum from k=0 to n of 1/k! * (n-k)^k.
We can round this down to bn=sum from k=0 to n of (n-k choose k). If we do that, we get bn=Fn+1, where Fn is the n'th Fibonacci number. (This is a well-known sum.) Or, in other words, about φn+1/√5, where φ is the golden ratio.
On the other hand, we can round it up to cn=sum from k=0 to n of (n-k multichoose k). If we do that, we get cn=2n-1 for n≥1 (while c0=1). (This is easy.)
So the original sequence an, being somewhere between these two, is evidently about exponential, with base somewhere inbetween φ and 2. Indeed, running the numbers, it is apparently about
1/(1+W(1))*1/W(1)n, or, equivalently, 1/(1+W(1))*eW(1)n
where W is the Lambert W function (i.e. W(1) is the unique real number such that 1/W(1)=eW(1)). Our base here, 1/W(1) (or eW(1)), is about 1.76322283...
So, that's kind of neat, but I have no idea why this might be. I don't really feel like thinking about it now, though, so I thought I'd just throw it out there and see if anyone knew anything about it...
-Harry
no subject
Date: 2012-01-10 05:39 pm (UTC)in an empyrical way.
In particular, was W(1) recognized by its first digits in the decimal expansion?
-Juan
no subject
Date: 2012-01-10 07:58 pm (UTC)no subject
Date: 2012-01-10 08:52 pm (UTC)the sequence A072597[n] in the OEIS.
Round( (1+W)^(-1) Exp[w n] (n-1)! )
gives the sequence
1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331,
23000296155, 486655768525, 11155073325918, 275364320099805,
7282929854486422, 205462851526617530, 6158705454187353312
that coincides upto n = 14.
For n = 18 A072597[n]= 6158705454187353297 the approximation differs only in
only in 15 units.
As they are integers, perhaps it can be computed
obtaining the complete asymptotic expansion.
-Juan