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When $n$ is odd, a cohomology of type Hochschild for $n$-ary partiallyassociative algebras has been defined in Gnedbaye's thesis. Unfortunately, thecohomology definition is not valid when $n$ is even. This fact is found againin the computations of the $n$-ary partially associative free algebra.  In this work, we define in a first time two approachs of an Hochschildcohomology for $n$-ary partially associative algebras. First by reducing thespace of cochains, secondly by using a graded version. Next we compute the free$n$-ary algebra, giving a basis of this algebra. At last we extend the notionof coalgebras to $n$-ary algebras.

n-ary associative algebras, cohomology, free algebras and coalgebras