[seqfan] Re: n-multisets of integers in (-n..n} adding to n

Max Alekseyev maxale at gmail.com
Sat Apr 14 04:35:57 CEST 2012

On Thu, Apr 12, 2012 at 3:55 PM, Robert Gerbicz
<robert.gerbicz at gmail.com> wrote:
> 2012. április 12. 18:06 Maximilian Hasler írta, <maximilian.hasler at gmail.com
>>:
>
>> On Thu, Apr 12, 2012 at 4:03 AM, David Scambler wrote:
>> > Count all n-multisets of integers in {-n, ..., n} such that the members
>> sum to n.
>>
>> This is the same than n-multisets of integers in [0..2n] such that the
>> members sum to n(n+1).
>>
>> So it might be interesting to have the more general function
>>
>> f( n,m,k ) = # of n-multisets in [0..m] whose elements sum up to k.
>>
>> This cannot obviously be coded as one simple table
>> (although for given n and m it will be zero for large enough k (> n*m)
>> which nevertheless would allow to pack all in one table or sequence)
>>
>> One could make one table f(n,m,k) for each n > 2
>>
>> or alternatively / additionnally only consider m = a*n
>> and make a table for
>>
>> a=1 <=> m=n (already there ?)
>> a=2 <=> m=2n (yours, the above)
>> a=3 <=> m=3n, etc.
>>
>>
>> Regards,
>>
>> Maximilian
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
> A tricky way:
> f(n,m,k)=polcoeff(prod(i=0,m,sum(j=0,n,x^(j*i*((m+1)*n+1)+j))),k*((m+1)*n+1)+n)

This can be further simplified -- the internal sum represents a
geometric progression.

Regards,
Max