Can anyone find a pattern?

[sql_lall]

Ok, i managed to use rs^a+rt^a+sr^a+st^a+tr^a+ts^a to show that (A+B+C)*P(a) - (AB+BC+CA)*P(a-1) + (ABC)*P(a-2) = P(a+1)
Where P(x) = A^x+B^x+C^x

From this i get:
p(1) = S
p(2) = S² - 2T
p(3) = S³ - 3ST + 3U
p(4) = S⁴ - 4S²T + 4SU + 2T²
p(5) = S⁵ - 5S³T + 5S²U + 5ST² - 5TU
p(6) = S⁶ - 6S⁴T + 6S³U + 9S²T² - 12STU - 2T³ + 3U²

Where
S = A+B+C
T = AB+BC+AC
U = ABC

Now, the pattern at the start is really easy. (powers of S increase by 1, coeff. increase by one)
Even then 2-5-9 infront of the S^xT² is just combinations stuff, but i'm not sure how to explain the # of terms in p(6). Previously, the number of terms in each p() increased by one, but then it increases by two suddenly . It then only increases by one in p(7). Weird

Anyway, if anyone can see a pattern, post away...