I defined a function in Mathematica online and I get this strange output. What does the triangle inside the octagon mean? Also, what is [n]? Also, why is there a y in the output when I haven't defined it?
f[k_,n_]=Sum[k!/(n!(n-k)!),{k,0,n}]
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2 Answers
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Well, I guess it is a shorthand for this elaborate function which I got by evaluating your expression in Mathematica on my desktop.
(1/n!)DifferenceRoot[
Function[{\[FormalY], \[FormalN]}, {-\[FormalY][\[FormalN]] + (5 +
2 \[FormalN]) \[FormalY][
1 + \[FormalN]] + (-10 -
6 \[FormalN] - \[FormalN]^2) \[FormalY][
2 + \[FormalN]] + (3 + \[FormalN]) \[FormalY][
3 + \[FormalN]] == 0, \[FormalY][1] == 2, \[FormalY][2] == 7/
2, \[FormalY][3] == 26/3}]][n]
answered Sep 23, 2021 at 19:07
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As noted in the comments, this is a DifferenceRoot object. If you click on the oblong, you get a few more details:
What this means is that the denominator $y(n)$ satisfies the relationships $$ (n+3) y(n+3) - (n^2 + 6n + 10) y(n+2) + (2n+5) y(n+1) - y(n) = 0 \\ y(1) = 2 \qquad y(2) = \frac{7}{2} \qquad y(3) = \frac{26}{3}. $$ These relations are sufficient to determine all of the coefficients. Usually this also means that Mathematica can't simplify the result any more than this.
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1$\begingroup$ Any ideas as to why it's represented as a circle with a triangle in the middle? $\endgroup$– JojoCommented Sep 24, 2021 at 7:34
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2$\begingroup$ @Joe: My best guess is that the triangle is supposed to be a delta (∆), which is a symbol for differences. People who work on ordinary difference equations or partial difference equations often abbreviate them as "O∆E"s and "P∆E"s to distinguish them from differential equations. $\endgroup$ Commented Sep 24, 2021 at 10:52
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1$\begingroup$ Oh I see that's interesting thank you. I've not seen them called O$\Delta$E before. $\endgroup$– JojoCommented Sep 24, 2021 at 13:57
\\FullFormat the end of the line and running it again $\endgroup$