Let $p$ be a prime with $p>3$, and let $a,b$ be two rational $p-$integers. In this paper we present general congruences for $sum_{k=0}^{p-1}binom akbinom{-1-a}kfrac p{k+b}pmod {p^2}$. For $n=0,1,2,ldots$ let $D_n$ and $b_n$ be Domb and Almkvist-Zudilin numbers, respectively. We also establish congruences for $$sum_{n=0}^{p-1}frac{D_n}{16^n},quad sum_{n=0}^{p-1}frac{D_n}{4^n}, quad sum_{n=0}^{p-1}frac{b_n}{(-3)^n},quad sum_{n=0}^{p-1}frac{b_n}{(-27)^n}pmod {p^2}$$ in terms of certain binary quadratic forms.
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