Hello Wolfram killers,
Sasha was near, but how to obtain these two integrals? Ok 500 is to much for calculating, but 200, ok?
f[r_] := Sum[(((-1)^n*(2*r - 2*n - 7)!!)/(2^n*n!*(r - 2*n - 1)!))*
x^(r - 2*n - 1), {n, 0, r/2}];
Nw := Transpose[Table[f[j], {i, 1}, {j, 5, 500, 1}]];
X1 = Integrate[Nw . Transpose[Nw], {x, -1, 1}];
Y1 = Integrate[D[Nw, {x, 2}] . Transpose[D[Nw, {x, 2}]], {x, -1, 1}];
MatrixForm[X1]
MatrixForm[Y1]
There is no need for it. Firstly,
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:
gives $$ \frac{2}{2 n+3}\ , $$ $$ \frac{12}{32 n^5+240 n^4+560 n^3+360 n^2-142 n-105}\ , $$ $$ -\frac{8}{32 n^5+400 n^4+1840 n^3+3800 n^2+3378 n+945}\ , $$ $$ \frac{2}{32 n^5+560 n^4+3760 n^3+12040 n^2+18258 n+10395}\ . $$