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Re: multinomials: a programming question
*To*: mathgroup at smc.vnet.net
*Subject*: [mg3961] Re: multinomials: a programming question
*From*: danl (Daniel Lichtblau)
*Date*: Mon, 13 May 1996 01:46:02 -0400
*Organization*: Wolfram Research, Inc.
*Sender*: owner-wri-mathgroup at wolfram.com
In article <4menll$jva at dragonfly.wolfram.com> Jack Goldberg
<jackgold at admin.lsa.umich.edu> writes:
> Hello Mma users,
>
> I am concerned about some limitations in the Sum command.
> Here is the example that motivated this message. Although
> Mma can easily expand (a+b+c)^n for any specific non-negative
> choice of integer n, I need to write the trinomial expansion
> out in terms of Sum. Something like this:
>
> tri[a_,b_,c_,n_] := Sum[Multinomial[i,j,k]*a^i*b^j*c*k, ???]
> where i+j+k = n. (Multinomial is a built-in function.)
>
> The problem is with the iterator(s). The condition i+j+k = n
> is causing me great difficulty. What would be nice is a solution
> that works in the general multinomial case, but perhaps that is asking
> too much - I would be happy for the trinomial expansion.
>
> The more general issue here is this: Many sums in mathematics (most
> more important than the above trivial problem) are indexed over more
> complicated sets than allowed by the syntax of Sum. For example,
> open any book on number theory - I opened my copy of An Introduction
> To The Theory of Numbers, I. Niven et. al. - and found the Mobius
> inversion formula, pg 194, part
> of which reads (subject to the limitation of my keyboard)
>
> The sum over all divisors d of n of the product of
> mu(d)*F(n/d) ...
>
> Instead of having to recast this remarkable formula into terms
> understandable by Sum, wouldn't it be nice to have the set over
> which the sum is taken be given as the iterator. Then my trinomial
> problem would be solved like this:
>
> tri[a_,b_,c_,n_] := Sum["as above", {i+j+k=n}]
>
> Your thoughts are more than welcome!
>
> Jack Goldberg
> University of Michigan
>
>
This is along the lines of what you want, for the simple case of
trinomials.
In[72]:= tri[a_,b_,c_,n_] := Sum[Multinomial[i,j,n-i-j]*a^i*b^j*c^(n-i-j),
{i, 0, n}, {j, 0, n-i}]
In[73]:= tri[a,b,c,4]
4 3 2 2 3 4 3 2
Out[73]= a + 4 a b + 6 a b + 4 a b + b + 4 a c + 12 a b c +
2 3 2 2 2 2 2 3 3
4
> 12 a b c + 4 b c + 6 a c + 12 a b c + 6 b c + 4 a c + 4 b c
+ c
For the more general case I believe the fragment below is correct.
In[114]:= multinom[n_Integer, x___] := Module[
{len=Length[{x}], jj, indices, xx={x}, expons, iterators},
indices = Array[jj, {len-1}];
expons = {Apply[Sequence,indices], n-Apply[Plus,indices]};
iterators = Table[{indices[[j]], 0, n-Apply[Plus, Take[indices,
j-1]]},
{j, len-1}];
Sum[Multinomial[Apply[Sequence,expons]] * Apply[Times, xx^expons],
Evaluate[Apply[Sequence, iterators]]]
]
In[115]:= multinom[3, w, x, y, z]
3 2 2 3 2 2 2
Out[115]= w + 3 w x + 3 w x + x + 3 w y + 6 w x y + 3 x y + 3 w y +
2 3 2 2 2
> 3 x y + y + 3 w z + 6 w x z + 3 x z + 6 w y z + 6 x y z + 3 y z
+
2 2 2 3
> 3 w z + 3 x z + 3 y z + z
Daniel Lichtblau
Wolfram Research, Inc.
danl at wolfram.com
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