Front-loaded Boxes

Six boxes labeled 1,2,,61,2, \ldots, 6 are arranged in a line. Seven identical balls are to be placed into the boxes such that for any 1k6,1 \leq k \leq 6, there are at least kk total balls amongst boxes 1,2,k.1,2 , \ldots k. How many different placements of the balls into the boxes are possible?

Details and assumptions

All 7 balls have to be placed.

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