Programmable Multinomial Expansion

[EngWiPy]

Hello,

How can we write a programmable multinomial expansion? I mean the summation over all non-negative integer indices, is somewhat difficult to program in Mathematica, at least for me. Is there any suggestions, please?


I saw this formula at the Mathematica official website, but I didn't know how to interpret it:

$$\left(a_1+a_2+\cdots+a_m\right)^n=\sum_{n_1,n_2,\dots,n_m=0}\delta_{n;n_1+n_2+\cdots+n_m}\left(n;n_1,n_2,\ldots,n_m\right)\prod_{k=1}^ma_k^{n_k}$$

what does it mean $$n_1,n_2,\ldots,n_m=0$$? and what is the delta function $$\delta_{n;n_1+n_2+\cdots+n_m}\left(n;n_1,n_2,\ldots,n_m\right)$$??

Thanks in advance

 

[mathman]

$$n_1,n_2,\ldots,n_m=0$$ means start the sum with all n_i=0.
$$\delta_{n;n_1+n_2+\cdots+n_m}\left(n;n_1,n_2,\ldots,n_m\right)$$ =1 when the sum of the n_i=n and =0 otherwise.

Note: For some reason I get an s where there should be ... In the checking before I submit it is correct?

 

[EngWiPy]

$$n_1,n_2,\ldots,n_m=0$$ means start the sum with all n_i=0.
$$\delta_{n;n_1+n_2+\cdots+n_m}\left(n;n_1,n_2,\ldots,n_m\right)$$ =1 when the sum of the n_i=n and =0 otherwise.

well, after we start all the indices from 0? Can you elaborate more, because I have no clear idea how to program this expansion in Mathematica.

Thanks in advance

 

[mathman]

I don't know anything about mathematica. Also (more important) the description you gave does not indicate what the upper limits of the summations for the m indices. In any case each n_i goes from 0 to its upper limit.

 

[EnumaElish]

See http://mathworld.wolfram.com/MultinomialSeries.html