examday

/TMUA

An integer $n$ is chosen uniformly at random from $1\le n<2^{18}$. It is given that exactly $64$ coefficients in the expansion of $(1+x)^n$ are odd, and that the coefficient of $x^n$ in $(1+x)^{3n}$ is even.

Let $T(n)$ be the number of odd coefficients in the expansion of $(1+x+x^2)^n$.

What is the probability that $T(n)$ is a perfect square?