Let $a(n,m)$ be an integer sequence such that $$a(n,m)=\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$$ Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ $$f(n)=n-2^{\ell(n)}$$ $$g(n)=[n+1\ne 2^k]\cdot(g(\left\lfloor\frac{n}{2}\right\rfloor)+1), g(0)=0$$ Here square brackets denote Iverson brackets, $f(n)$ is the same as $n$ without the most significant bit and $g(n)$ is A063250, i.e., number of binary right-rotations to reach fixed point.
Let $b(n,m)$ be an integer sequence such that $$b(n,m)=m(b(f(n),m)+[g(n)>0]\cdot b(f(n)+2^{g(n)-1},m)), b(0,m)=1$$ Let $s(n,m)$ be an integer sequence such that $$s(n,m)=\sum\limits_{j=0}^{2^n-1}b(j,m)$$ I conjecture that $$s(n,m)=a(n+1,m)$$ Is there a way to prove it?
