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COMMENTS
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Because the inverse binomial transform of A000045 and A113405 is individually the same as the original sequence up to a sign flip of each second term, the same is true for their difference here. (The inverse binomial transform is a linear transform.)
The sequence and its higher order differences in subsequent rows has zeros on the main diagonal:
0, 1, 1, 1, 1, 1, 1, -1, -7,-23, -59, -139, -311, -677, -1443, -3031
1, 0, 0, 0, 0, 0, -2, -6,-16,-36, -80, -172, -366, -766, -1588, -3264
-1, 0, 0, 0, 0,-2, -4,-10,-20,-44, -92, -194, -400, -822, -1676
1, 0, 0, 0,-2,-2, -6,-10,-24,-48,-102, -206, -422, -854, -1732
-1, 0, 0,-2, 0,-4, -4,-14,-24,-54,-104, -216, -432, -878, -1764
1, 0,-2, 2,-4, 0,-10,-10,-30,-50,-112, -216, -446, -886, -1790,
-1,-2, 4,-6, 4,-10, 0,-20,-20,-62,-104, -230, -440, -904, -1792
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