identity involving symmetric group

Hi. Here's a puzzle.

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Let j>=1 be an integer. Let n_1,n_2,...,n_j be j positive integers with
the following two properties:

1) the average of the n_i is 2
2) none of the n_i are 2

[e.g. j=5 and the integers could be 1,1,1,1,6]

Consider the product

j(j-n_1)(j-n_1-n_2)(j-n_1-n_2-n_3)...(j-n_1-n_2-...-n_j)

(note that the last term in the product is just -j)

Well, this product is unfair because it relies on the order of the n_i.
So now sum this product over all j! permutations of the n's.

[i.e. form sum_{g in Symm(j)} j(j-n_{g1})(j-n_{g1}-n_{g2})...(-j) ]

Is this sum always zero?

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Remark 1): if one lets some of the n_i be 2 the sum sometimes isn't 0.

Remark 2): this is actually a *highly disguised* question about mod p
modular forms! [more precisely, about their expansions around supersingular
points of the modular curve] It's not an idle question at all. A grad student
of mine has run up against it and it's got to the stage where he can't
do it, I can't do it, and we both just want it out of the way.

Thanks for any help,

Kevin Buzzard