Simplex number calculation (all dimensions) in python

How can I calculate simplex numbers in all dimensions?

Example:

dimension:     2|      3|      4|
simpl(1)       1|      1|      1|
simpl(2)       3|      4|      5|
simpl(3)       6|     10|     15|
simpl(4)      10|     20|     35|
simpl(5)      15|     35|     70|

First I considered about a recursive implementation (in python):

def simplex(n, dim):
    if dim == 1:
        return n
    else:
        i = 1
        ret = 0
        while i <= n:
            ret += simplex(i, dim-1)
            i+=1
        return ret

The algorithm works, but because of the recursion its really slow, especially if your dimension number is 5 or higher.

You should use a for loop rather than a while loop to solve the problem.
This is as you're just re-writing a for loop.
Using a for loop you can change the loop to a comprehension, and take the sum of it.
This allows you to get the code:

def simplex(n, dim):
    if dim == 1:
        return n
    dim -= 1
    return sum(simplex(i, dim) for i in range(1, n + 1))

However this can be simplified.
When dim is the bullet point the function is:

• \$n\$

• \$\Sigma_{i = 1}^{n} i\$

\$= \frac{n(n + 1)}{2}\$

• \$\Sigma_{i = 1}^{n} (\Sigma_{j = 1}^{i} j)\$

\$= \frac{n(n + 1)(n + 2)}{6}\$

• \$\Sigma_{i = 1}^{n} (\Sigma_{j = 1}^{i} (\Sigma_{k = 1}^{j} k)))\$

\$= \frac{n(n + 1)(n + 2)(n + 3)}{24}\$

• \$\Sigma_{i = 1}^{n} (\Sigma_{j = 1}^{i} (\Sigma_{k = 1}^{j} (\Sigma_{l = 1}^{k} l))))\$

\$= \frac{n(n + 1)(n + 2)(n + 3)(n + 4)}{120}\$

Since we're getting a pattern, we know that the top part of the fraction is:

\$\Pi_{i = 0}^{dim - 1} n + i\$

And if we use oeis to find the denominator it says it's probably the factorial sequence.
And so it's likely:

\$\frac{\Pi_{i = 0}^{dim - 1} n + i}{dim!}\$

This is actually really easy to write in Python.
And so you can get a significant speed up.

from functools import reduce
from math import factorial
from operator import mul

def simplex(n, dim):
    return reduce(mul, (n + i for i in range(dim))) // factorial(dim)

StackExchange Code Review Q#142777, answer score: 3