Consider the binary partitions of $2n$ in powers of $2$, denoted by $b(2n)$, with the generating function
$$\sum_{n\geq0}b(2n)\,x^n=\frac1{1-x}\prod_{k\geq0}\frac1{1-x^{2^n}}.$$
A result of De Bruijn shows the asymptotic growth
$$\log b(2n)\sim \frac1{\log 4}\log^2\left(\frac{n}{\log n}\right).$$
Suppose $t(3n)$ denote the ternary partitions of $3n$ in powers of $3$, with a similar generating function
$$\sum_{n\geq0}t(3n)\,x^n=\frac1{1-x}\prod_{k\geq0}\frac1{1-x^{3^n}}.$$
QUESTION. What is the asymptotic growth of $t(3n)$?
Best Answer
It's right in the beginning of De Bruijin's paper. More generally,
$\log p(rn) \sim \frac 1 {2 \log r} \log^2 \frac n {\log n}$
where $p(rn)$ is the number of partitions of $rn$ into powers of $r$.
The result is attributed to Kurt Mahler's paper "On a special functional equation", published in Journ. London Math. Soc. in 1940.