# Plot with sum and binomial command

A few days ago I asked about a problem with plotting a sum. You advise me to use Evaluate option, and that helped me very. But now I have a very similar problem, but i can't find the mistake. I have a equation: $$\sum_{i=0}^x {x\choose l}\cdot l^{10}\cdot(-1)^{(x-l)}$$ I want to draw this for $$x \in [0,10]$$ I used this command:

Plot[Evaluate[Sum[Binomial[x, l]*l^50*(-1)^(x - l), {l, 0, x}]], {x, 1,2}]


There are no errors, but also no graph. If I will expand the equation for $x=1$ I have: $${1\choose 0 }\cdot 0^{10}\cdot (-1)^{1-0} + {1 \choose 1}\cdot 1^{10}\cdot (-1)^{1-1} = 1$$

and for example for $x=5$ it is: $${5\choose 0 }\cdot 0^{10}\cdot (-1)^{5-0} + {5 \choose 1}\cdot 1^{10}\cdot (-1)^{5-1} + {5\choose 2 }\cdot 2^{10}\cdot (-1)^{5-2} + {5 \choose 3}\cdot 3^{10}\cdot (-1)^{5-3} + {5\choose 4 }\cdot 4^{10}\cdot (-1)^{5-4} + {5\choose 5 }\cdot 5^{10}\cdot (-1)^{5-5} = 5 103 000$$

So as you can see, there equation can be evaluated, so why I can't see any plotted points?

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You have terms like (-1)^(-0.5) which is a complex number. You need to restrict x to integers, or take the Abs[ ] or plot over a range where you don't get complex-valued numbers. –  bill s Dec 6 '13 at 1:31
@bill Is any other way to restrict x to integers than DiscretePlot? –  Ziva Dec 6 '13 at 1:46

## Revised version

It seems you don't want to use Plot, because this creates a continuous plot in the region. You want to draw the sum for integer values of x. It takes a good amount of time, but your sum can be evaluated analytically

s = Sum[Binomial[x, l]*l^50*(-1)^(x - l), {l, 0, x}]

(* (-(-1)^x)*x*
HypergeometricPFQ[{2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 - x},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1}, 1] *)


and now you can create the data you like

Table[s, {x, 1, 5}]

(* {1, 1125899906842622, 717897984314152868242380, \
1267647728636285389485789180120, 88811503724190361952542576750515000} *)


or plot it using ListPlot or ListLinePlot (I use Log here, because the values increase very fast)

ListLinePlot[Log[Table[s, {x, 1, 10}]]]


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Do I correctly understand that in the solution which you suggested this sum is evaluated from l=0 to 1 when x=1 and again from l=0 to 2 when x=2? And does 'Re@' mean repeat? –  Ziva Dec 6 '13 at 1:50
First mistake which I made was that I didn't restriced x to integers. –  Ziva Dec 6 '13 at 1:53
@Ziva, right, it was not obvious from your question that you only want integers for x. Plot samples the region between 1 and 2 very fine and the sum is evaluated for x=1, 1.00001, 1.00002, ... which looks wrong. If you only want integer solutions, the Plot is not the way to go. –  halirutan Dec 6 '13 at 1:55
@Ziva Let me update my answer. –  halirutan Dec 6 '13 at 1:56
Thank you for your help. I have one more question for this a = 21 s = Sum[Binomial[x, j]*j^a*(-1)^(x - j), {j, 0, x}] Table[s, {x, 1, 2}] this is working wrong, because it count that for x = 1 s=-10 905180. For a=23 or a =11 it works good. Why for a=21 it is working wrong? –  Ziva Dec 6 '13 at 16:57

I am confused by the test which has exponent 10 and the code which has exponent 50 so have dealt with both:

f[x_, n_] := Total[Binomial[x, #] #^n (-1)^(x - #) & /@ Range[0, x]]


Visualizing:

GraphicsRow[
ListLogPlot[Table[f[x, #], {x, 1, 5}], Filling -> Axis,
PlotStyle -> {Red, PointSize[0.02]}, FillingStyle -> Thick,
AxesOrigin -> {0, 1},
PlotLabel ->
Row[{"n=", #}, BaseStyle -> {FontFamily -> "Calibri", 16}],
ImageSize -> 300] & /@ {10, 50}]


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