How to sum products of vector elements in a set of vectors?

Question

Let $a$ be a set of vectors such that $a = \{q^2, q^3, q^4\, ...\}$, where $q^k$ is a vector of length $2^k$, and of the form $q^k = \{q[1], q[2], ..., q[k]\}.$ I want to evaluate a sum like $\sum\limits_{k=2}^{N-1} \sum\limits_{h>k}^N \sum\limits_{i=1}^{n_k} q_{i}^k q_{i}^h$, where $n_k$ is the length of the vector $q^k$, and $N$ is the cardinality of $a$. This involves taking the product of components of vectors in seperate elements of the set. I made an attempt and got a nasty set of errors.

a = Array[q, 4]
b = Array[r, 8]
c = Array[s, 16]
z = {a, b, c}
OverlapV[v_] := 
Subscript[\[Lambda], overlap] Sum[With[{l = Length[v[[k]]]}, Sum[Sum[
  v[[k]][[i]] v[[h]][[i]], {i, 1, l}, {h, i + 1, Length[v]}], {h, 
  i + 1, Length[v]}]], {k, 1, Length[[v[k]]] - 1}]

I managed to accomplish something similar for the term $\sum\limits_{k=2}^{N} \sum\limits_{i = 1}^{n_k - 1} \sum\limits_{j > i}^{n_k} q_{i}^k q_{j}^k$ after some help in a similar posted question with this code:

One[x_] := With[{l = Length[x]},
Sum[Sum[x[[i]] x[[j]], {i, 1, l}, {j, i + 1, l}], {i, 1, l - 1}]]
OneV[v_] := 
Subscript[\[Lambda], one] Sum[F[v[[k]]], {k, 1, Length[v]}]

which gives the output: This is what I am trying to output for the first term. I am new to Mathematica, so I apologize if the code is messy or not idiomatic.

Answer 1

You may use Indexed and Sum.

With

a = Indexed[q, #] & /@ Range[1, #] & /@ Range[1, 4]

Then

Sum[a[[k, i]]*a[[h, i]], 
  {k, 2, Length[a] - 1}, 
  {h, k + 1, Length[a]}, 
  {i, 1, Length[a[[k]]]}]

or when entered with Esc+sumt+Esc

IMHO, the above bit of code demonstrates one of the best features of Mathematica: Your code can be written in mathematical notation.

Hope this helps.

Answer 2

I'm not sure if I understood what you wanted correctly, but it seems you wanted something like this:

a = Array[q, 4];
b = Array[r, 8];
c = Array[s, 16];
z = {a, b, c};

Total[Dot @@@ (PadRight[#, {2, Min[Length /@ #]}] & /@ Subsets[z, {2}])]
   q[1] r[1] + q[2] r[2] + q[3] r[3] + q[4] r[4] + q[1] s[1] + r[1] s[1] + q[2] s[2] +
   r[2] s[2] + q[3] s[3] + r[3] s[3] + q[4] s[4] + r[4] s[4] + r[5] s[5] + r[6] s[6] +
   r[7] s[7] + r[8] s[8]

As I understood it, you want all possible pairs of your vectors in $a$ (thus, Subsets[]), take the dot product of the members of each pair (with truncation as needed, hence PadRight[]), and then sum that all up.