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Let $\\{B_n(x)\\}$ and $\\{E_n(x)\\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\\sum_{k=0}^{p-1}\\binom ak\\binom{-1-a}k\\equiv (-1)^{\\langle a\\rangle_p}+ p^2t(t+1)E_{p-3}(-a)\\pmod{p^3}$$ and for $a\\not\\equiv -\\frac 12\\pmod p$, $$\\sum_{k=0}^{p-1}\\binom ak\\binom{-1-a}k\\frac 1{2k+1}\\equiv \\frac{1+2t}{1+2a} +p^2\\frac{t(t+1)}{1+2a}B_{p-2}(-a)\\pmod{p^3},$$ where $\\langle a\\rangle_p\\in\\{0,1,\\ldots,p-1\\}$ satisfying $a\\equiv \\langle a\\rangle_p\\pmod p$ and $t=(a-\\langle a\\rangle_p)/p$. 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