Let $f(x)$ be continuous positive value function on$[0,1]$, and $a_k:=\int _0^1 x^k f(x)dx$ .Then $A:=(a_{i+j})_{0\leq i,j\leq n-1}$ is invertible.
$A$ is Hermitian matrix, so, if $ det (a_{i+j})_{0\leq i,j\leq k} >0$, $A$ has only positive eigen values and $detA>0$. But I can't prove it and I don't know whether $A$ is positive definite.
To prove positive definitivity, not that for any $0\neq y=(y_0,y_1,\dots,y_{n-1})\in \mathbb R^n$, $$y^\top Ay=\int_0^1\big(y_0+y_1x+y_2x^2+\dots+y_{n-1}x^{n-1}\big)^2f(x)\,dx>0.$$