Row Alignment in Grid

Question

I've been working on creating tables of differences in MAA similar to the image below, but with a bit more detail:

I've written a function that will take an anonymous function, a range of values and some optional arguments to create the table:

differenceTable[f_: Function[{g}, g], range_List, n_: 3, diff_: 3, p_: 2] :=
Module[
{a = N[f /@ range, n]}, 
Grid[
{{"x"} ~ Join ~ {"f[x]"} ~ Join ~ (Table[Superscript[\[CapitalDelta], k], {k, 1, diff}]),
{Column[range, Center, 1]} ~ Join ~ (Column[#, Center, 1] & /@ (SetPrecision[#, p] & /@ diffList[a, diff + 1]))},
Spacings -> 2, 
Dividers -> {{False, True}, {False, True}}
]
]

with,

diffList[x_List, n_] := 
  Module[{d = Differences[x]}, {x} ~ Join ~diffList[d, n - 1]];
diffList[x_List, 1] := {x}

For example:

differenceTable[Sin[#] &, Range[10, 90, 10] Degree]

gives

Which is great. The problem I run into is that when the function values and the x values take on non-standard formatting (fractional notation and exponentials specifially) the spacing between rows is thrown off.

For example:

differenceTable[Sin[#] &, Range[Pi/18, Pi/2, Pi/18]]

gives:

I'm not sure how to fix this without eliminating the taper in the table, which I would like to keep. Does anyone know of a fix for this?

Answer 1

I've started from the beginning, but with Grid with number of rows equal to 2*Length[range] and SpanFromAbove inside:

     diffT[f_, range_, columns_] := 
      Module[{n = Length@range, RF, dif, grid, form},
      RF = Append[Evaluate@Riffle[#, SpanFromAbove], SpanFromAbove] &;
      form = NumberForm[N@#, {5, 4}] &;
      dif = NestList[Differences, f@# & /@ range, columns];
      grid = PadRight[#, 2 n, ""] & /@ (PadLeft[#, n + Length@#/2,""] & /@ (RF /@ (Map[form, dif, {2}])));
      Grid[
      Transpose[{RF@range, ##} & @@ grid]
      , ItemSize -> 4]
       ]

You can easily add all Your arguments or Headings.

It defenitely need to be optimized. Thanks in advance for remarks.