<s>[INST] I found a correlation between n^x and x! can anyone explain it? This is the example to numbers to the 3rd.
Here is the same thing just with numbers to the 4th
Well I was messing around with numbers I made this discovery, with the correlation between n^x and x!. What is the reason for this? Is it easily explainable?
In the sheets every column is found from the differences of two numbers in sequence in the last column
 [/INST] The binomial theorem says
$$
(a+b)^x = a^x + xa^{x-1}b + \frac{x(x-1)}2 a^{x-2} b^2 + \frac{x(x-1)(x-2)} 6 a^{x-3} b^3 + \cdots.
$$
The differences in the column to the right of the one that lists values of $a^x$ are
$$
(a+1)^x - a^x = \left( a^x + x a^{x-1} + \frac{x(x-1)} 2a^{x-2} + \cdots \right) - a^x.
$$
Since $a^x$ cancels, you have $xa^{x-1}$ as the highest-degree term in the resulting polynomial function of $a.$
Then for the next row, for the same reason you get $x(x-1)x^{x-2}$ as the highest-degree term.
And next you get $x(x-1)(x-2)a^{x-3},$ and so on until finally you have $x(x-1)(x-2)\cdots 1 a^{x-x}.$
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