I have found some number sequence $c_n = 1+b_n$ for $n \ge 0$, where $b_n = $ [A307977(n)][1].
I am trying to solve the following recurrence relation for the prime counting function:

$$\forall n \ge 3: \pi(n) = \frac{1}{n-2} \sum_{k=0}^{n-1} \pi(k) c_{n-1-k}$$

in terms of the sequence $c_n$, where $\pi(n)=0$ for $n<2$ and $\pi(2)=1$. (I can prove that the recurrence formula holds, but I can not solve the recurrence formula for $\pi(n)$ on the left and $c_k$-s only on the right as suggested in the SageMath code:

Here is for $n \ge 3$ the sequence $(n-2)! \cdot \pi(n)$ computed with [SageMath][2]:

    3 c_0
    4 c_0^2 + c_1
    5 c_0^3 + 3*c_0*c_1 + 2*c_2
    6 c_0^4 + 6*c_0^2*c_1 + 3*c_1^2 + 8*c_0*c_2 + 6*c_3
    7 c_0^5 + 10*c_0^3*c_1 + 15*c_0*c_1^2 + 20*c_0^2*c_2 + 20*c_1*c_2 + 30*c_0*c_3 + 24*c_4
    8 c_0^6 + 15*c_0^4*c_1 + 45*c_0^2*c_1^2 + 40*c_0^3*c_2 + 15*c_1^3 + 120*c_0*c_1*c_2 + 90*c_0^2*c_3 + 40*c_2^2 + 90*c_1*c_3 + 144*c_0*c_4 + 120*c_5
    9 c_0^7 + 21*c_0^5*c_1 + 105*c_0^3*c_1^2 + 70*c_0^4*c_2 + 105*c_0*c_1^3 + 420*c_0^2*c_1*c_2 + 210*c_0^3*c_3 + 210*c_1^2*c_2 + 280*c_0*c_2^2 + 630*c_0*c_1*c_3 + 504*c_0^2*c_4 + 420*c_2*c_3 + 504*c_1*c_4 + 840*c_0*c_5 + 720*c_6

**Q1) Is it possible to give an explicit formula for $\pi(n)$ in terms of $c_k$ only?**

Q2) Are there other known recursive formulas for the prime counting function in the literature?

(I tried to search for some of these coefficients in OEIS but I can not see the general pattern, if there is one, for all coefficients of these polynomials in the $c_n$.)

Thanks for your help!


  [1]: https://oeis.org/A307977
  [2]: https://sagecell.sagemath.org/?z=eJydk09vgjAYxu8kfIfGXd5CVYrLDovs7GExcTsui4HSShOtphTix1-xKDDZlowT7fu8D7_3D3WqYWIm2Pd8L-cCCQGGrBMaRfjZ95B9NDeVVqisDmCCQJlie9LywEFiJI4aSSQV0qnacaBkHVJ8s1qt_rAKLkbbkwTlvNSIV5Pn_BgDlTxF1nLRWb7NluYFJeg1rTRX5p1rycs3qXaw2RDV5hdW0NKs26vcXuVWXKdG1hwKYvC8cKEHi6UM5K2SNcoZO3IhJJP2GyVcIzOpSq4NxCTCgwrZlduRZ9l_yYUVtCP5gVw05GLw9Xta38vONg0uIDRuSB4tCbaErtjsjK-vlzYPNa4KXZR2UMNR0jmoaYwDaGYKNMzOHzKkn7g32imd3q-Kwt2e7Ed8u3S7BTcAzipte8V4p5cCqWXcHnoGURdOkpE4dTd8X_KR4KCquvlF2HYSlkbbrcc46IP8Wt63pV6Q-DZ-12xFXFdJr2J7ECkzRy3TveNwRxjUj_EXg08B5A==&lang=sage&interacts=eJyLjgUAARUAuQ==