Binomial Summations 3

Expression is symmetric. Making a $13 \times 13$ grid, we list out all the possible ordered pairs of $(i, \ j)$ . As expression is symmetric, both sides of the diagonal from $( \ 0, \ 0 )$ to $(12, \ 12)$ have equal sum. So we just need to find both side's sum and then half it. This is done by evaluating the non inequality constrained sum and then subtracting the sum of elements forming the diagonals i.e. where $i \ = \ j$ . Doing so and dividing by $2$ we get $12 \times \ 2^{12} \ = \boxed{49152}$