How can I prove the following?
1-x+x^2+x^5-x^7-x^12+x^15-x^22-x^26+x^35-x^40+...
= Product[(1 - x^(8 i - 7)) (1 + x^(8 i - 6)) (1 + x^(8 i - 5)) (1 + x^(8 i - 4)) (1 + x^(8 i - 3)) (1 + x^(8 i - 2)) (1 - x^(8 i - 1)) (1 - x^(8 i))] Where the product is from i=1 to infinity.$$1-x+x^2+x^5-x^7-x^{12}+x^{15}-x^{22}-x^{26}+x^{35}-x^{40}+\dots \\= \prod_{i=1}^{\infty} [(1 - x^{8 i - 7}) (1 + x^{8 i - 6}) (1 + x^{8 i - 5}) (1 + x^{8 i - 4}) (1 + x^{8 i - 3}) (1 + x^{8 i - 2}) (1 - x^{8 i - 1}) (1 - x^{8 i})]$$
It doesn't seem to follow from the Triple Product formula and I haven't been able to come up with a combinatorial proof.
