In how many ways can I sum integers from $1$ to $N-1$ to obtain $N$? I'm looking for the exact formula for $f(N) =$ number of ways to sum $1, 2, ..., N-1$ to obtain $N$. 
$N$ is an integer $> 0$. Integers $1, 2, ..., N-1$ can be used $0$ or 1 time as an element in the sum.
For example:
$f(6) = 3$. There are 3 ways to obtain 6 from integers less than $6$:
$1+2+3$
$1+5$
$2+4$

The power series for
$$
\prod_{k=1}^\infty\left(1+x^k\right)
$$
which is equal to
$$
\prod_{k=1}^\infty\frac1{1-x^{2k-1}}
$$
is the generating function for your number plus $1$. The beginning of the power series is
$$
\small1+x+x^2+2x^3+2x^4+3x^5+4x^6+5x^7+6x^8+8x^9+10x^{10}+12x^{11}+15x^{12}+\dots
$$
