Supercongruences involving Motzkin numbers and central trinomial coefficients

Abstract Let M n and T n denote the n th Motzkin number and the n th central trinomial coefficient respectively. We prove that for any prime p 5, equation* aligned\\-22pt &₊=₀^p-1Mₖ² (p3) (2-6p) p², \\ &₊=₀^p-1kMₖ² (p3) (9p-1) p², \\ &₊=₀^p-1TₖMₖ 43 (p3) +p6 (1-9 (p3) ) p², \\-6pt aligned equation* where (-) is the Legendre symbol. These results confirm three supercongruences conjectured by Z. -W. Sun in 2010.