infinite q series

1032 days ago by pub

How to compute something about

\prod_{n=1}^\infty (1-q^{2n})^2(1-q^{3n})^3(1-q^{4n})^{-4}(1-q^{5n})^{-5n}
R.<q> = PowerSeriesRing(ZZ) # Always work to precision 20: q = q + O(q^20) 
       
prod(1-q^(2*n) for n in (1..20)) 
        
1 - q^2 - q^4 + q^10 + q^14 + O(q^21)
1 - q^2 - q^4 + q^10 + q^14 + O(q^21)
prod(1-q^(2*n) for n in (1..20))^2 * prod(1-q^(3*n) for n in (1..20))^3 
        
1 - 2*q^2 - 3*q^3 - q^4 + 6*q^5 + 2*q^6 + 3*q^7 + q^8 - q^9 + 2*q^10 -
13*q^11 - 2*q^12 - 11*q^13 + 16*q^15 - 2*q^16 + 5*q^17 - 9*q^18 +
16*q^19 + 15*q^20 + O(q^21)
1 - 2*q^2 - 3*q^3 - q^4 + 6*q^5 + 2*q^6 + 3*q^7 + q^8 - q^9 + 2*q^10 - 13*q^11 - 2*q^12 - 11*q^13 + 16*q^15 - 2*q^16 + 5*q^17 - 9*q^18 + 16*q^19 + 15*q^20 + O(q^21)