How to use currying to write a tail recursive function for summing factorials given a certain bound

Question

I am very confused about how can i use currying to write a tail recursive function for summing factorials within a certain bound or summing the bounds i.e. given [1,5] i should be getting 1!+2!+3!+4!+5! and for summing bounds 1+2+3+4+5. How can i construct two elegant functions using currying.

My code currently looks like this:

factorial[n_] := (loopfact[n, acc_] := If[n == 0, acc, loopfact[n - 1, acc*n]];
Return@loopfact[n, 1]);

sum[a_, b_, acc_] := Block[{x, ac}, x = a; ac = acc;
loop[x_, ac_] := If[x > b, ac, loop[x + 1, ac + factorial[x]]];
loop[x, ac]]

sum[0, 5, 0] (* using the function above *)
(* 154 *)

sum[a_, b_, acc_] := Module[{x, ac}, x = a; ac = acc;
loop[x_, ac_] := If[x > b, ac, loop[x + 1, ac + # &@x]];
loop[x, ac]]

sum[0, 50, 0] (* using the second function definition *)
(* 1275 *)

Also could someone kindly explain to me how currying works in Mathematica. Thanks !

Answer 1

If you are more interested in currying than tail-recursion, this is not the answer you want. However, I don't think currying, and certainly not your approach to currying, is very suited to Mathematica. (It will be interesting if Leonid Shifrin or someone else of his level of expertise will step in a show that I'm wrong).

If tail-recursive is really the focus of your question, then I recommend a Scheme style approach with tail-recursive helper functions. This is very clean stylistically (no If, but only pattern matching on arguments).

First the factorial.

g[1, val_] = val;
g[k_ /; k > 1, val_] := g[k - 1, k val]
f[1] = 1;
f[k_Integer /; k > 1] := g[k - 1, k]

Now the summation.

s[a_, a_, val_] := val + f[a]
s[a_, b_, val_] := s[a, b - 1, val + f[b]]
sum[a_Integer /; a > 0, b_Integer] /; b >= a := s[a, b, 0]

sum[1, 5] == Plus @@ Factorial /@ Range[1, 5]

True

Edge case.

sum[10, 10] == Factorial[10]

True