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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] 
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    I don't know why you don't sim$p$ly mark your co$d$e but it would be great if you can just click the code button when you are entering your Mathe$m$atica code. See the questio$n$ edit for the difference.2011-11-04

1 Answers 1

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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}\ . $

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    no it is not the good solution. 200x200 doesn't work on all cores because it is integrating. The solution which Sasha suggests is good but it is not working $f$or derivative. First we solve the simple inte$g$ral and then use parallel table, in that case all cores are working. But the problem is because I can not obtain for derivative. Take a look http://math.stackexchange.com/questions/78996/how-to-solve-integral-in-mathematica2011-11-04