Abstract. A famous result due to Grothendieck asserts that every continuous linear operator from ℓ 1 to ℓ 2 is absolutely (1, 1)-summing. If n ≥ 2, however, it is very simple to prove that every continuous n-linear operator from ℓ 1 × · · · × ℓ 1 to ℓ 2 is absolutely (1; 1, ..., 1)-summing, and even absolutely 2 n ; 1, ..., 1 -summing. In this note we deal with the following problem:Given a positive integer n ≥ 2, what is the best constant g n > 0 so that every n-linear operator from ℓ 1 × · · · × ℓ 1 to ℓ 2 is absolutely (g n ; 1, ..., 1)-summing?We prove that g n ≤ 2 n+1 and also obtain an optimal improvement of previous recent results (due to Heinz Juenk et al , Geraldo Botelho et al and Dumitru Popa) on inclusion theorems for absolutely summing multilinear operators.