Hankel-type determinants for some combinatorial sequences

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 $n=0,1,2,\dots$, we show that $6^{-n}\mathrm{\mid }{f}_{i+j}{\mathrm{\mid }}_{0\le i,j\le n}$ and $12^{-n}|D_{i+j}|{0\le i,j\le n}$arepositiveoddintegers,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.