Prove that the number of circuits with bounded fan-in (of 2) of size $s$ is at most $s^{O(s)}$.

Give the best explicit bound you can get.

I know that for a circuit of size $s$ in order to generate a string to represent it I need for every edge, which I have at most $s$ of those, to represent from which vertice it start and in which vertice it ends and for that I need $2s log(s)$ bits.

In addition I need another 2 bit for every vertice to mark whether it is an input, an OR AND NOT Gate.

So overall I need $O(s log(s))$ bits to represent a circuit, and there are $2^{O(s log(s))} = s^{O(s)}$ circuits like that.

Am I in the right direction or am I missing something? And is $2^{3(s log(s))} = 8s^{O(s)}$ the bound?

Thanks:)