How many partitions do I have?

Mathematica, 11 bytes

PartitionsP

Explanation

¯\_(ツ)_/¯

Pyth, 3 bytes

l./

Try it here! or Try a test Suite.

The answer took much longer to format than writing the code itself :P.

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How?

Pyth is the right tool for the job.

l./   Full program with implicit input.

 ./   Integer partitions. Return all sorted lists of positive integers that add to the input.
l     Length.
      Implicitly output the result.

Python 2, 85 83 bytes

-2 bytes thanks to @notjagan

lambda n:n<1or sum(sum(i*((n-k)%i<1)for i in range(1,n+1))*p(k)for k in range(n))/n

Try it online!

Using recursive formula from OEIS A000041.