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There are infinitely many polynomials of order 6 that will give these as the first six 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 simplest solution, based on a polynomial of order 5 is:

U(n) = (32*n^5 - 440*n^4 + 2360*n^3 - 6010*n^2 + 7133*n - 3070)/5.

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Q: What is the rule pattern for 1 4 19 94 469 2344?
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