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Up to the multiplcative factor traced back from $q^{1/24}$, there is the following expansion:
sage: PSR.<q> = PowerSeriesRing( QQ, default_prec=20 )
sage: f = qexp_eta( PSR, 20 )
sage: f
1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + O(q^20)
sage: f(q) / f(q^2) / f(q^4)
1 - q - q^3 + 2*q^4 - 2*q^5 + q^6 - 2*q^7 + 5*q^8 - 5*q^9 + 3*q^10 - 5*q^11 + 10*q^12 - 10*q^13 + 7*q^14 - 11*q^15 + 20*q^16 - 20*q^17 + 15*q^18 - 22*q^19 + O(q^20)
Note: ?qexp_eta describes the used function.
Note: See also ?EtaProduct - EtaProduct is already implemented for the case where the product has "an integer power of $q$ cumulated from the many factors".
Here is the link: etaproducts.html
