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I'm looking for an efficient way of computing sums of the type $$D_{jk} = \sum_{i=1}^n A_{ij} B_{ik} C_{i}$$ for large numerical matrices $A$, $B$ and $C$.

Here is a slow solution:

DD = Table[Sum[AA[[i, j]] BB[[i, k]] CC[[i]], {i, 1, n}], {j, 1, m}, {k, 1, m}]

What's a fast way of doing it?

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n = 4;
m = 3;
aa = Array[a, {n, m}];
bb = Array[b, {n, m}];
cc = Array[c, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], {i, 1, n}], {j, 1, m}, {k, 1, m}];

You can Transpose aa and Dot it with bb cc:

Transpose[aa].(bb cc) == dd

True

Alternatively, you can Transpose aa cc and Dot it with bb:

Transpose[aa cc].bb ==dd

True

Comparison with Table/Sum combination:

SeedRandom[1]
n = 100;
m = 50;
aa = RandomReal[1, {n, m}];
bb = RandomReal[1, {n, m}];
cc = RandomReal[1, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], {i, 1, n}], {j, 1, m}, {k, 1,   m}]; //
   RepeatedTiming // First

0.417

ee1 = Transpose[aa].( bb cc) ; // RepeatedTiming // First

0.000051

ee2 = Transpose[aa cc].bb ; // RepeatedTiming // First

0.000045

dd == ee1 == ee2

True

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