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There are two major problems in answering this question.

The first problem is that there are infinitely polynomials of order 7 that will give these as the first seven numbers and any one of these could be "the" rule. There are also non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.

The second problem is that you have not specified where, in the sequence the missing number should have been.

If the missing number was the last, the simplest polynomial rule of order 6 ist(n) = (-36*n^6 + 854*n^5 - 7945*n^4 + 36680*n^3 - 87299*n^2 + 99386*n - 39600)/120 for n = 1, 2, 3, ... and, accordingly the next number is -396.

If missing number was the first, the simplest polynomial rule of order 6 is different and the first number is 330.

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