Reformulation

We define recursively the energy E(v1 ... vn) of a sequence of
integers v1 ... vn as E(v1) = v1 for a sequence of length 1 and as
the minimum for i∈1..n of E(v1 ... vi) + E(vi+1 ... vn) + Σ vi otherwise.

The problem is to compute the energy of a given sequence v1 ... vn.
The dynamic recursive question is almost given in the subject.

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q( v1 ... vk) is "what is the energy to build the sequence v1 ... vk"
and we have the recursive formula:

       q( v1 ... vk )  = min_i q( v1 ... vi ) + q(vi+1 ... vk )  

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We could also look at the boolean version (not that useful here)

q( v1 ... vk) is "is E the energy to build the sequence v1 ... vk"
and we have the recursive formula:

       q( v1 ... vk, E )  = ∃ E1, E2,i such that E1+E2=E and  q( v1 ... vi,E1 ) + q(vi+1 ... vk ,E2 )  
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We need to notice that among all possible arguments to q we will only
use sequences of the form L_i ... L_j where L_1 ... L_N is the input
sequence.  Therefore we can rewrite the dynamic question into

q(i,j) = what is the energy E( L_i ... L_{j-1}) ?

and see that

q(i,j) = Σ_{i≤k<j} L_k + min_k q(i,k)+q(k,j)

which can very easily be turned into code...