Why does the number of ways that $n$ can be summed with at least one $1$ equal the partition function for $n-1$?

Question

For some reason I was counting the number of partitions of $n$ that have at least one $1$ as an addend. The beginning sequence for these numbers, starting with $n=1$, is $\{1, 1, 2, 3, 5, 7, 11, 15, 22,\dots\}$. This is as far as I went, but so far this is exactly the sequence for the partitions-counting function. I'm curious as to whether or not this sequence continues as such and, if so, why does my function, $p_1(n)$ equal the partition function, $p(n-1)$?