Translate mathematic formula to Javascript

Question

I have this formula:

f(x)=(x^0+x^0+x^1+x^101+x^2+x^202)^n

which is basically probabilities when rolling n dice with faces showing values:

{0,0,1,101,2,202}

How do I translate that to JavaScript?

I know I can do recursive function to get every single results and add them in an array of results, but anything over 9 dice becomes very long.

EDIT: Note that x is not a variable. It is some fancy mathematique symbol that represent I forgot what about probabilities.

Answer 1

The x in your question is a formal variable, and you're asking for a procedure to compute powers of formal polynomials in that variable. Here's a quick implementation. Warning: I haven't executed it, and in any case it lacks polish. But I hope you get the idea.

function polynomial_unit() {
  return [1]; // Represents 1 = 1x^0
}

function polynomial_mult(x, y) {
  var z = Array.apply(null, new Array((x.length + y.length - 1))).map(Number.prototype.valueOf,0);
  for(var i = 0; i < x.length; i++) {
    for(var j = 0; j < y.length; j++) {
      z[i+j] += x[i]*y[j];
    }
  }
  return z;
}

function generic_exp(base, exponent, mult, unit) {
  var t = unit;
  while (exponent > 1) {
    if (exponent%2 == 1) t = mult(t, base);
    base = mult(base, base);
    exponent = Math.floor(exponent/2);
  }
  return mult(t, base);
}

function Wildhorn(n) {
  var f = Array.apply(null, new Array(203)).map(Number.prototype.valueOf,0);
  f[0] = 2; f[1] = 1; f[2] = 1; f[101] = 1; f[202] = 1;
  return generic_exp(f, n, polynomial_mult, polynomial_unit());
}