There is no need for it. Firstly,
f[r_] = FullSimplify[
Sum[(((-1)^n*(2*r - 2*n - 7)!!)/(2^n*n!*(r - 2*n - 1)!))*
x^(r - 2*n - 1), {n, 0, r/2}], r > 0 && r \[Element] Integers]
simplifies to
$$
f(r)=-\frac{\sqrt{\pi } (-1)^r 2^{r-3} x^{r-1} \, _2\tilde{F}_1\left(\frac{1-r}{2},1-\frac{r}{2};\frac{7}{2}-r;\frac{1}{x^2}\right)}{\Gamma (r)}.
$$
Caculations show that only a few diagonals have non-zero elements:
$$\tiny
\left(
\begin{array}{ccccccccccc}
\frac{4}{315} & 0 & -\frac{8}{10395} & 0 & \frac{2}{45045} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{4}{3465} & 0 & -\frac{8}{45045} & 0 & \frac{2}{135135} & 0 & 0 & 0 & 0 & 0 \\
-\frac{8}{10395} & 0 & \frac{4}{15015} & 0 & -\frac{8}{135135} & 0 & \frac{2}{328185} & 0 & 0 & 0 & 0 \\
0 & -\frac{8}{45045} & 0 & \frac{4}{45045} & 0 & -\frac{8}{328185} & 0 & \frac{2}{692835} & 0 & 0 & 0 \\
\frac{2}{45045} & 0 & -\frac{8}{135135} & 0 & \frac{4}{109395} & 0 & -\frac{8}{692835} & 0 & \frac{2}{1322685} & 0 & 0 \\
0 & \frac{2}{135135} & 0 & -\frac{8}{328185} & 0 & \frac{4}{230945} & 0 & -\frac{8}{1322685} & 0 & \frac{2}{2340135} & 0 \\
0 & 0 & \frac{2}{328185} & 0 & -\frac{8}{692835} & 0 & \frac{4}{440895} & 0 & -\frac{8}{2340135} & 0 & \frac{2}{3900225} \\
0 & 0 & 0 & \frac{2}{692835} & 0 & -\frac{8}{1322685} & 0 & \frac{4}{780045} & 0 & -\frac{8}{3900225} & 0 \\
0 & 0 & 0 & 0 & \frac{2}{1322685} & 0 & -\frac{8}{2340135} & 0 & \frac{4}{1300075} & 0 & -\frac{8}{6194475} \\
0 & 0 & 0 & 0 & 0 & \frac{2}{2340135} & 0 & -\frac{8}{3900225} & 0 & \frac{4}{2064825} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{2}{3900225} & 0 & -\frac{8}{6194475} & 0 & \frac{4}{3151575}
\end{array}
\right),$$
$$
\left(
\begin{array}{ccccccccccc}
\frac{2}{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{2}{7} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \frac{2}{9} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{2}{11} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \frac{2}{13} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{2}{15} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{2}{17} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{2}{19} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{2}{21} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{2}{23} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{2}{25}.
\end{array}
\right)$$
And those diagonals satisfy simple formulas:
FindSequenceFunction[Diagonal[Y1], n] // Simplify
FindSequenceFunction[Diagonal[X1, 0], n] // Simplify
FindSequenceFunction[Diagonal[X1, 2], n] // Simplify
FindSequenceFunction[Diagonal[X1, 4], n] // Simplify
gives
$$
\frac{2}{2 n+3}\ ,
$$
$$
\frac{12}{32 n^5+240 n^4+560 n^3+360 n^2-142 n-105}\ ,
$$
$$
-\frac{8}{32 n^5+400 n^4+1840 n^3+3800 n^2+3378 n+945}\ ,
$$
$$
\frac{2}{32 n^5+560 n^4+3760 n^3+12040 n^2+18258 n+10395}\ .
$$