Let $m\geqslant1$ be a fixed integer.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then we have an integer sequence given by \begin{align} a(0)&=1\\ a(n)& = ma\left(\frac{n-2^{f(n)}}{2}\right)+(-1)^{n}a\left(\left\lfloor\frac{2n-2^{f(n)}}{2}\right\rfloor\right) \end{align} I conjecture that $$a\left(\frac{4^n-1}{3}\right)=m^{n}n!L_{n}\left(\frac{1}{m}\right)$$ where $L_{n}(x)$ is Laguerre polynomials.
I also conjecture that $a(2^p(2^q-1))$ relates with $(p+1)$-Stirling numbers of the second kind, but I have problems with offset.
Is there a way to prove it?
