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Prove $\sum_{j=0}^{n} q^{j^{2}}\binom{n}{j}_{q^{2}}$ generates the self-conjugate partitions with part at most $n$.
Prove $\sum_{j=0}^{n} q^{j^{2}}\binom{n}{j}_{q^{2}}$ generates the self-conjugate partitions with part at most $n$, and that it equals $(1+q)(1+q^{3})\cdot\cdot\cdot(1+q^{2n-1})$.
For the first part, ...
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How to prove it? (one of the Rogers-Ramanujan identities)
Prove the following identity (one of the Rogers-Ramanujan identities) on formal power series by interpreting each side as a generating function for partitions:
$$1+\sum_{k\geq1}\frac{z^k}{(1-z)(1-z^2)...
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