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 Aτ 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 Hτ associated to the symbol τ into a sum of indecomposable operators of low rank. A basis of Aτ 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.