An identity for the ratio of two partial Bell polynomials

Question

Let $B_{\ell,m}(x_1,x_2,\dotsc,x_{\ell-m+1})$ denote the Bell polynomials of the second kind (or say, partial Bell polynomials, (exponential) partial Bell partition polynomials). I knew that the identity \begin{equation*}\label{0Y0}\tag{o-o} \frac{B_{\ell+2m+2,\ell}\bigl(1, 0, 1, 0, 9, 0,\dotsc,[(2m-1)!!]^2, 0, [(2m+1)!!]^2\bigr)} {B_{\ell+2m,\ell}\bigl(1, 0, 1, 0, 9, 0,\dotsc,[(2m-3)!!]^2, 0, [(2m-1)!!]^2\bigr)} =(\ell+2m)^2 \end{equation*} is valid for $\ell=1,2$ and $m\in\mathbb{N}$, but not necessarily valid for $\ell\ge3$ (as the counterexample by Peter Taylor at https://mathoverflow.net/a/456761 shows).

What is the general expression of the ratio \begin{equation*} R(\ell,m)=\frac{B_{\ell+2m+2,\ell}\bigl(1, 0, 1, 0, 9, 0, 225,0,\dotsc,[(2m-1)!!]^2, 0, [(2m+1)!!]^2\bigr)} {B_{\ell+2m,\ell}\bigl(1, 0, 1, 0, 9, 0, 225,0,\dotsc,[(2m-3)!!]^2, 0, [(2m-1)!!]^2\bigr)} \end{equation*} for $\ell\ge3$ and $m\in\mathbb{N}$?

Thank you very much for your attention.

References

1. Ch. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.
2. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974; available online at https://doi.org/10.1007/978-94-010-2196-8.
3. Feng Qi, Taylor's series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi, Demonstratio Mathematica 55 (2022), no. 1, 710--736; available online at https://doi.org/10.1515/dema-2022-0157.
4. Feng Qi, Explicit formulas for partial Bell polynomials, Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine, and series representations of powers of Pi, Research Square (2021), available online at https://doi.org/10.21203/rs.3.rs-959177/v3.
5. Feng Qi, Gradimir V. Milovanovic, and Dongkyu Lim, Specific values of partial Bell polynomials and series expansions for real powers of functions and for composite functions, Filomat 37 (2023), no. 28, 9469--9485; available online at https://doi.org/10.2298/FIL2328469Q.
6. Feng Qi and Peter Taylor, Series expansions for powers of sinc function and closed-form expressions for specific partial Bell polynomials, Applicable Analysis and Discrete Mathematics 18 (2024), no. 1, in press; available online at https://doi.org/10.2298/AADM230902020Q.

Answer 1

Counterexample: consider $\ell = 3$, $m = 1$. The LHS is $$\frac{B_{7,3}({\color{red} 1}, {\color{green} 0}, {\color{blue} 1}, 0, 9)} {B_{5,3}({\color{red} 1}, {\color{green} 0}, {\color{blue} 1})} = \frac{105 \cdot {\color{green} 0}^2 \cdot {\color{blue} 1} + 70 \cdot {\color{red} 1} \cdot {\color{blue} 1}^2 + 105 \cdot {\color{red} 1} \cdot {\color{green} 0} \cdot 0 + 21 \cdot {\color{red} 1}^2 \cdot 9} {15 \cdot {\color{red} 1} \cdot {\color{green} 0}^2 + 10 \cdot {\color{red} 1}^2 \cdot {\color{blue} 1}} = \frac{259} {10}$$

Answer 2

For $\ell,m\in\mathbb{N}$, the first three values of the ratio $R(\ell,m)$ are \begin{align*} R(1,m)&=(1+2m)^2,\\ R(2,m)&=(2+2m)^2,\\ R(3,m)&=(3+2m)^2\frac{\sum_{k=0}^{m+1}\frac{1}{(2k+1)^2}}{\sum_{k=0}^{m}\frac{1}{(2k+1)^2}}. \end{align*} This implies that the general expression of the ratio $R(\ell,m)$ shouldn't be of a simple form.