Construct the $n$-tuple Cartesian product of the ternary set $X_n=\{0,1,2\}\times\cdots\times\{0,1,2\}=\{0,1,2\}^n$. Define its subset $W_n$ according to the rule (here $y=(y_1,\dots,y_n)$ is made use of) $$W_n=\{y\in X_n: y_1\leq1, y_1+y_2\leq2,\dots,y_1+\cdots+y_{n-1}\leq n-1, y_1+\cdots+y_n=n\}.$$ Introduce the one-variable polynomial $$P_n(x)=\sum_{y\in W_n}x^{\prod_{j=1}^n\binom{2}{y_j}}.$$

QUESTION.If $C_n$ stands for the Catalan numbers, is this true? $$\frac{dP_n}{dx}(1)=C_{n+1}.$$

**Examples.** $P_1(x)=x^2, P_2(x)=x^4+x, P_3(x)=x^8+3x^2, P_4(x)=x^{16}+6x^4+2x$.