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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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I don't have a name for it, but you might be interested that $x=\frac{(9+8+7+6)^2-(9^2+8^2+7^2+6^2)}{2}$, which reduces the number of computations when there are lots of digits. –  Ross Millikan Aug 22 '11 at 23:43
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I would be very surprised if such a thing had a name! –  Mariano Suárez-Alvarez Aug 22 '11 at 23:44
    
@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 :/ –  Templar Aug 22 '11 at 23:52
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Not specific to sums of digits, but symmetric polynomials are up this alley. Does this help?

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