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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

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Let  $q<50$  be a prime number. Call a sequence of polynomials  $P_0(x), P_1(x), P_2(x), ..., P_{q^2}(x)$  *tasty* if it satisfies the following conditions: 



- $P_i$  has degree  $i$  for each  $i$  (where we consider constant polynomials, including the  $0$  polynomial, to have degree  $0$ )
- The coefficients of  $P_i$  are integers between  $0$  and  $q-1$  for each  $i$ .
- For any  $0\le i,j\le q^2$ , the polynomial  $P_i(P_j(x)) - P_j(P_i(x))$  has all its coefficients divisible by  $q$ .


As  $q$  varies over all such prime numbers, determine the total number of tasty sequences of polynomials.

*Proposed by Vincent Huang*

Present the answer in LaTex format: \boxed{Your answer}
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