<s>[INST] Sequence and polynomial power relationship I recently encountered this relationship between polynomial powers and a certain associated sequence and I am seeking any help or idea that might answer why the relationship is true.
Let $P(x)$ be a polynomial, say for instance $P(x)=1+3x+2x^2$. Consider the consecutive powers of $P(x)$ and arrange the numerical coefficients in order of appearance. For the given $P(x)$ we have for up to fifth power:
$P(x)^1=1+3x+2x^2$
$P(x)^2=1 + 6 x + 13 x^2 + 12 x^3 + 4 x^4$
$P(x)^3=1 + 9 x + 33 x^2 + 63 x^3 + 66 x^4 + 36 x^5 + 8 x^6$
$P(x)^4=1 + 12 x + 62 x^2 + 180 x^3 + 321 x^4 + 360 x^5 + 248 x^6 + 96 x^7 + 
 16 x^8$
$P(x)^5=1 + 15 x + 100 x^2 + 390 x^3 + 985 x^4 + 1683 x^5 + 1970 x^6 + 
 1560 x^7 + 800 x^8 + 240 x^9 + 32 x^{10}$.
Also, consider the sequence defined by
$a_{m,n}=a_{(m-1),n}+3a_{(m-1),(n-1)}+2a_{(m-1),(n-2)}$ with $a_{1,1}=1,a_{1,2}=3$ and $a_{1,3}=2$.
Observe that the sequence above completely determines the entries in the expansion of the polynomial power. For instance the number $63$ in the third power of $P(x)$ is equal to $63=12+3(13)+2(6)$.
I am wondering why is this TRUE. Thanks for your help and suggestion.
 [/INST] The sequence you encountered is used to construct a particular extension of the $a-b$-based triangle. In particular, your sequence generates the $1-3-2$  triangle. In general, for $a_0a_1\ldots a_{r-1}$-based triangle, the entry in the $m^{th}$ row and $n^{th}$ column which is exactly the term $a_{mn}$ in your recurrence is given by the numerical coefficient of $x^n$ in the expansion of $(a_0+a_1x+\ldots+a_{r-1}x^{r-1})^m$.
Hence, $a_{mn}$ is the coefficient of $x^n$ in the expansion of $[P(x)]^m$.
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