g = Table[
  ListPlot[
    Table[{k, Binomial[k + r - 1, r - 1] (r/(10 + r))^r (1 - r/(10 + r))^k}, {k, 0, 25}],
    PlotRange -> 0.118, 
    Filling -> Axis,
    PlotStyle -> PointSize[0.02], 
    FillingStyle -> Directive[Hue[0.6], Thickness[0.01], Opacity[0.3]],
    Epilog -> Inset[Style["\!\(\*StyleBox[\"r\",\nFontSlant->\"Italic\"]\) = " <> ToString[r], 16], {23, 0.08}], 
    Prolog -> {Opacity[0.5], Green, Thin, Line[{{10 - Sqrt[(10 (10 + r))/r], .1}, {10 + Sqrt[(10 (10 + r))/r], .1}}], 
               Dashed, Thick, Orange, Line[{{10, 0}, {10, 1}}]},
    ImageSize -> 200,
    LabelStyle -> Directive[Larger],
    AxesLabel -> {"\!\(\*StyleBox[\"k\",\nFontSlant->\"Italic\"]\)"},
    AxesOrigin -> {-1, 0}
    ], 
  {r, {1, 2, 3, 4, 5, 10, 20, 40}}
  ]
Export["c:/users/stpasha/documents/negbinomial.gif", g, "DisplayDurations" -> {.5, .5, .5, .5, .5, .5, .5, .15}, "TransparentColor" -> White, ImageSize -> 250]

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