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1

Migrated from here Mathematica is quite likely to be slow at generating permutations. Here is a Java-based approach. It is based on Java reloader and this nice Java code, which I slightly extended with a getNextMultiple method. So, load the Java reloader first (run the code from that post). Then, execute the following: JCompileLoad @ " import ...


3

Without expanding Binomial into Gamma functions, you can also see that the result is correct based on the following true statement: SeriesCoefficient[(1 + x)^n, {x, 0, k}, Assumptions -> k >= 0] (* ==> Binomial[n, k] *) This is the binomial expansion, valid in particular for n = -1. But that case leads to the alternative expression ...


6

Looked up the rules on this. This is how it works. If $n$ and $r$ are negative integers, there is a symmetry relation $\binom{n}{r}=\binom{n}{n-r}$ and now the limit is used. But now $\binom{n}{r}=\binom{-1}{0}$ from above. Hence the above limit is, where $n=-1$ and $r=0$ is n = -1; r = 0; Limit[ Gamma[n + t + 1]/(Gamma[r + 1] Gamma[n + t - r ...


2

If you use: SB[n_?NumericQ, r_?NumericQ] in your definition things work as you expect. Otherwise, SB is evaluated symbolically and that will take forever...


0

Clear["Global`*"] k = 6.; SB[n_, r_] := Sum[Binomial[r Binomial[2 k, 2]/2, i] Binomial[ Binomial[n, 2] - r Binomial[2 k, 2]/2, r Binomial[k, 2] + r - i], {i, r Binomial[k, 2] + r/2, r Binomial[k, 2] + r}] SB[# k, #] & /@ Range[100] // Timing ListLogPlot[%[[2]]] way 2 Clear["Global`*"] sum[r_] := Sum[(Gamma[1 + 33 r] Gamma[1 - 36 r + 18 r^2])/( ...



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