In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Ap\'ery numbers. For any nonnegative integer $n$, define \begin{ gather*}f_n:=\sum_{ k=0}^n\binom nk^3,\ D_n:=\sum_{ k=0}^n\binom nk^2\binom{ 2k}k\binom{ 2(n-k)}{ n-k}, b_n:=\sum_{ k=0}^n\binom nk^2\binom{ n+k}k,\ A_n:=\sum_{ k=0}^n\binom nk^2\binom{ n+k}k^2. \end{ gather*} For $n=0,1,2,\ldots$, we show that $6^{ -n}|f_{ i+j}|_{ 0\leq i,j\leq n}$ and $12^{ -n}|D_{ i+j}|_{ 0\le i,j\le n}$ are positive odd integers, and $10^{ -n}|b_{ i+j}|_{ 0\leq i,j\leq n}$ and $24^{ -n}|A_{ i+j}|_{ 0\leq i,j\leq n}$ are always integers.
Categories: math.CO