I am trying to get an estimate for the induced 2-norm condition number $\kappa_{2}(M)$ of this matrix $M$: $$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} \frac{x^{n-j}}{(n-j)!}\:{\rm{d}}x,$$ which is symmetric positive definite of size $n\times n$. The above link gives the elements of the inverse. We also know that its determinant is the reciprocal of this OEIS sequence.

I want to obtain something like $\kappa_{2}(M) = \Omega(f(n))$ for some to-be-determined $f(n)$.

From numerics, it seems that $\kappa_2$ grows very fast. But I am not sure what might be an analytic way to obtain the asymptotic for $\lambda_{\max}/\lambda_{\min}$.

I am trying to get an estimate for the induced 2-norm condition number $\kappa_{2}(M)$ of this matrix $M$: $$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} \frac{x^{n-j}}{(n-j)!}\:{\rm{d}}x,$$ which is symmetric positive definite of size $n\times n$. The above link gives the elements of the inverse. We also know that its determinant is the reciprocal of this OEIS sequence.

I want to obtain something like $\kappa_{2}(M) = \Omega(f(n))$ for some to-be-determined $f(n)$.

From numerics, it seems that $\kappa_2$ grows very fast with respect to $n$. But I am not sure what might be an analytic way to obtain the asymptotic for $\lambda_{\max}/\lambda_{\min}$.