<s>[INST] Determine in How many ways $N!$ can be expressed as sum of consecutive numbers Determine in How many ways $N!$ can be expressed as sum of consecutive numbers.For example for $3!$ can be as $1+2+3$ only $1$ ways.
Please suggest an elegant approach.
 [/INST] Let's deal with odd length and even length sequences separately.
A number $N$ is the sum of an odd number of consecutive integers, centered at $m$, whenever $m|N$ and $N/m$ is odd.
A number $N$ is the sum of an even number of consecutive integers, centered at $m+1/2$, whenever $2m+1|N$.
Now let $N = 2^k M$, where $M$ is odd. For each factor $m|M$ we get one odd-length sequence centered at $2^km$, and one even-length sequence centered at $m+1/2$.
Example: $N = 6 = 2^1 3$, so we get two odd-length sequences (centered at $2$ and $6$) and two even-length sequences (centered at $.5$ and $1.5$): $$6 = 1 + 2 + 3 = 6 = -5 -4 -3 -2 -1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 0 + 1 + 2 + 3.$$
In order to obtain a formula for the number of representations of $n!$, all we need to do is to figure out how many odd divisors $n!$ has. I leave that to the reader.
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