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Lassalle's sequence is defined by the recurrence $A_1:=1$ and for $n\geq2$, $$A_n=(-1)^{n-1}C_n + (-1)^{n- 1}\sum_{j=1}^{n-1}(-1)^j\binom{2n - 1}{2j - 1}A_jC_{n - j}$$ where $C_k=\frac1{k+1}\binom{2k}k$ are the Catalan numbers. The sequence $A_n$ is found on OEIS together with related descriptions.

QUESTION. Is it true that $A_n$ is log-convex, i.e. $A_{n+1}A_{n-1}-A_n^2\geq0\,$?

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  • $\begingroup$ I assume you checked the first few $n$. For the small list in OEIS, it looks as if $\log(A_{n+1}A_{n-1}-A_n^2)$ is growing fairly rapidly, very roughly like $\log(n!)$. $\endgroup$
    – Joe Silverman
    Commented Mar 3, 2021 at 21:24
  • $\begingroup$ More experiments: Let $$B_n=\frac{A_n+1}{A_n}-\frac{A_n}{A_{n-1}}$$, so you want to prove that $B_n\ge0$. Experimentally, it looks as if $$\lim_{n\to\infty} B_{n+1}-B_n = \beta\quad\text{for some $\beta\approx2.1795$.} $$ $\endgroup$
    – Joe Silverman
    Commented Mar 3, 2021 at 21:30
  • $\begingroup$ According to a comment on the OEIS link (by Sergei N. Gladkovskii) $$\sum_{k\ge0} A_k\frac{x^{2k+2}}{(2k+2)!} = -\log u(x),$$ for $ u(x):= \frac{J_1(2x)}x$. This function satisfies an even simpler linear ODE than the Bessel function, namely $x\ddot u+3\dot u+4xu=0$. (But composing with $\log$ produces a nonlinear quadratic ODE, as suggested by the Cauchy product in the inductive definition). $\endgroup$
    – Pietro Majer
    Commented Mar 3, 2021 at 21:59
  • $\begingroup$ @JoeSilverman and Pietro: thank you both for interesting comments. $\endgroup$
    – T. Amdeberhan
    Commented Mar 3, 2021 at 22:32

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