Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant (Q344123)

The author considers the following numerical sequence defined by Somos: NEWLINE\[NEWLINE g_0=1, \hskip 1cm g_n=n\,g_{n-1}^2, \hskip 5mm n=1,2,3,\dots, NEWLINE\]NEWLINE and obtain the following results.NEWLINENEWLINEWhen \(n\to\infty\), NEWLINE\[NEWLINE g_n\sim{\sigma^{2^n}\over n}\exp\left[\sum_{k=1}^\infty{\alpha_k\over(n+\beta_k)^{2k-1}}\right] NEWLINE\]NEWLINE and NEWLINE\[NEWLINE g_n\sim{\sigma^{2^n}\over n}\exp\left[1+\sum_{k=1}^\infty{\lambda_k\over(n+\mu_k)^{2k-1}}\right], NEWLINE\]NEWLINE where \(\sigma\) is the Somos' recurrence constant and the coefficients \(\alpha_k\), \(\beta_k\), \(\lambda_k\) and \(\mu_k\) may be obtained from a non-linear recurrence.NEWLINENEWLINEFor any natural number \(n\), NEWLINE\[NEWLINE {\sigma^{2^n}\over n}\exp\left[-{2\over n+a}\right]\leq g_n\leq {\sigma^{2^n}\over n}\exp\left[-{2\over n+b}\right] NEWLINE\]NEWLINE and NEWLINE\[NEWLINE {\sigma^{2^n}\over n}\left[1-{2\over n+\lambda}\right]\leq g_n\leq {\sigma^{2^n}\over n}\left[1-{2\over n+\mu}\right], NEWLINE\]NEWLINE with the best possible constants \(a\), \(b\), \(\lambda\) and \(\mu\).