I'm not totally sure this fully answers your questions, but such relations are known for odd values of $k$: $$P_{2k+1}=\frac{1}{2^{2k+2}(2k+2)} \sum_{q=0}^k \binom{2k+2}{2q} (2-2^{2q})~ B_{2q} ~\left[(8P_1+1)^{k+1-q}-1\right]$$ where the $B_{2q}$'s are Bernoulli numbers (see Faulhaber's formula on Wikipedia).

I'm not totally sure this fully answers your questions, but such relations are known for odd values of $k$: $$P_{2h+1}=\frac{1}{2^{2h+2}(2h+2)} \sum_{q=0}^h \binom{2h+2}{2q} (2-2^{2q})~ B_{2q} ~\left[(8P_1+1)^{h+1-q}-1\right]$$ where the $B_{2q}$'s are Bernoulli numbers (see Faulhaber's formula on Wikipedia).

I'm not totally sure this fully answers your questions, but such relations are known for odd values of $k$: $$P_{2k+1}=\frac{1}{2^{2k+2}(2k+2)} \sum_{q=0}^k \binom{2k+2}{2q} (2-2^{2q})~ B_{2q} ~\left[(8P_1+1)^{k+1-q}-1\right]$$ where the $B_2q$'s are Bernoulli numbers (see Faulhaber's formula on Wikipedia).

I'm not totally sure this fully answers your questions, but such relations are known for odd values of $k$: $$P_{2k+1}=\frac{1}{2^{2k+2}(2k+2)} \sum_{q=0}^k \binom{2k+2}{2q} (2-2^{2q})~ B_{2q} ~\left[(8P_1+1)^{k+1-q}-1\right]$$ where the $B_{2q}$'s are Bernoulli numbers (see Faulhaber's formula on Wikipedia).