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Consider the following sequence of even polynomials $f_k:[0,1]\to\mathbb R$ , $$ \begin{cases} \,f_k(x)\,= \frac{(1-x^2)^2}{2}\,f_{k-1}''(x) \quad \textrm{for } k\geq 2\\ f_1(x)=x^2 \end{cases}$$

I am interested in the values $\phi_k:=f_{k}(0)=\frac{1}{2}f_{k-1}''(0)\,$. The first ones are $\phi_{1}=0$, $\phi_{2}=1$, $\phi_3=-2$, $\phi_4=10$, $\dots$

I would like to show that $\phi_k$ is $\geq0$ for $k$ even and $\leq0$ for $k$ odd. Moreover I would like to show $$\frac{|\phi_{2k}|}{(2k)!} \,\geq\, \max\Big(\frac{|\phi_{2k-1}|}{(2k-1)!},\,\frac{|\phi_{2k+1}|}{(2k+1)!}\Big) .$$

Any possible strategy?

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    $\begingroup$ Take a look at oeis.org/A005799 $\endgroup$ Commented Oct 1 at 17:17
  • $\begingroup$ I computed also $\phi_5,\phi_6,\phi_7$ : it looks exactly the sequence in your link, but with alternating signs $\endgroup$ Commented Oct 1 at 23:53
  • $\begingroup$ @jjagmath how did you see that my sequence was A005799? $\endgroup$ Commented Oct 4 at 15:13
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    $\begingroup$ I calculated the first 20 values and looked for it in the OEIS page. $\endgroup$ Commented Oct 4 at 16:37

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