We study the decomposition of a multi-symmetric tensor $T$ as a sum of powers of product of linear forms in correlation with the decomposition of its dual $T^*$ as a weighted sum of evaluations. We use the properties of the associated Artinian Gorenstein Algebra $A_\tau$ to compute the decomposition of its dual $T^*$ which is defined via a formal power series $τ$. We use the low rank decomposition of the Hankel operator $H_\tau$ associated to the symbol $\tau$ into a sum of indecomposable operators of low rank. A basis of $A_\tau$ is chosen such that the multiplication by some variables is possible. We compute the sub-coordinates of the evaluation points and their weights using the eigen-structure of multiplication matrices. The new algorithm that we propose works for small rank. We give a theoretical generalized approach of the method in n dimensional space. We show a numerical example of the decomposition of a multi-linear tensor of rank 3 in 3 dimensional space.