If you set $\alpha=1$ and impose the condition that $P_{2i}=P_{2i+1}$, then you get the sequence $P_0,P_2,P_4,... = 1,1/2,0,1/4,-1/4,0,1/4,1/8,-3/8,-1/8,1/8,0,...$. Define $Q_{2i}=P_{2i}-P_{2i-2}$. The pattern of these differences is simpler: $Q_2,Q_4,Q_6,...=-1/2,-1/2,1/4,-1/2,1/4,-1/8,-1/2,1/4,...,(-1/2)^{b(i)}$ where $b(i)$ is the number of $1$s in the binary expansion of $i$. To prove this, one can check that $Q_{2k}=Q_{4k}=-2Q_{4k+2}$ using $P_{2k}=2P_{4k+2}$.

Since there are arbitrarily large numbers of low binary weight, the differences $Q_{2i}$ do not converge to $0$ so the terms $P_i$ do not converge to $0$, so the sum does not converge and can't be normalized to $1$. Nevertheless, for other values of $\alpha$ I expect that you can produce a convergent series this way involving the binary weight function $b$.