Convolution with modified indices

Question

I want to perform convolution for a sequence but instead of having the indices cycle from the ends periodically, I want to apply a function, specifically Abs, to remap the indices.

Using ListConvolve, what parameters and/or modifications to the inputs could allow this output?

For example, here is my specific case. My input sequences will have equal length. Using a summation, I can obtain the results I desire, but this runs in $O(n^2)$.

For an arbitrary sequence of length 5, this would be the convolution using summation.

Table[Sum[Subscript[a, j] Subscript[b, Abs[k - j] + 1], {j, 5}], {k, 5}]

Answer 1

How about

k = Table[Subscript[a, j] , {j, 5}];    
l = Table[Subscript[b, 1 + Abs[j]] , {j, -4, 4}];

ListCorrelate[k, l] // Reverse // Column