Manipulate which creates and deletes Manipulators dynamically

Question

I want to plot a function in a Manipulate in which the number of terms varies and a Manipulator for the coefficient of each term is created or deleted as the terms increase and decrease.

For example, when I have three terms, I should get the equivalent of

Manipulate[
  Plot[a0 + a1 x + a2 x^2, {x, 0, 10}], 
  {a0, 1, 10}, 
  {a1, 1, 10}, 
  {a2, 1, 10}]

I want to let the number of terms vary and but still control each coefficient with a Manipulator. I tried to do it in the following way:

Manipulate[
  Plot[Total[ToExpression["a" <> ToString[#]] x^# & /@ Range[0, n]], {x, 0, 10}],
  {n, 1, 10, 1},
  Dynamic[Sequence @@ ({ToExpression["a" <> ToString[#]], 1, 10} & /@ Range[1, n])]]

It did not work.

Can anybody suggest what I should do to make it work?

Answer 1

Here's an idea

DynamicModule[{coef},
 _coef = 0;
 Manipulate[
  Column[{
    Grid@Partition[#, 2, 2, 1, {}] &[
      Array[
       Manipulator[Dynamic[coef[#]], {-4, 4}, 
         Appearance -> "Labeled"] &,
        order + 1, 0]
      ] // Dynamic,
    Dynamic@Plot[
      Sum[coef[i] \[FormalX]^i, {i, 0, order}], {\[FormalX], 0, 10}, 
      ImageSize -> Medium, PlotRange -> {-ran, ran}]
    }], 
  {{order, 1, "Order"}, 0, 10, 1, Appearance -> "Labeled"},
  {{ran, 10, "Plot Range"}, 0.1, 100, Appearance -> "Labeled"}]
 ]

Answer 2

Consider this:

    Manipulate[
     foo = Total@Map[#*x^# &, Range[0, n]];
     Plot[foo, {x, 0, 3}],
     Grid[
      {{
        Slider[Dynamic@n, {0, 10, 1}],
        Dynamic[foo, {n, 1, 10, 1}]
        }}
      ],
     TrackedSymbols -> n]

Could be further improved, this was only a quick shot.