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I'm looking for something like this and I want to know how it's called; I'm pretty sure there is a term for it.

I will show an example: Let's say we take the number 9876. $x=9\cdot8+9\cdot7+9\cdot6+8\cdot7+8\cdot6+7\cdot6$ You can see I multiply each pair of digits and then add.

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    @RossMillikan yeah I just got it from there just didn't multiply by 2 and thought it would be nice to remember, but it would be easier to remember if I know name of it, but maybe it doesn't exist :/2011-08-22

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Not specific to sums of digits, but symmetric polynomials are up this alley. Does this help?