find the first power of $t$ with a non-positive coefficient

edit

You can use power series as follows:

sage: q=4 ; m=33 ; k=31
sage: R.<t> = PowerSeriesRing(QQ)
sage: f = ((1-t^q)^(m-k) * (1-t^2)^m) / ((1-t)^(m-k+1) * (1-t^(2*q))^m)
sage: f
1 + 3*t - 27*t^2 - 89*t^3 + 343*t^4 + 1269*t^5 - 2701*t^6 - 11567*t^7 + 14569*t^8 + 75707*t^9 - 55699*t^10 - 379649*t^11 + 147471*t^12 + 1525661*t^13 - 225781*t^14 - 5106855*t^15 - 91515*t^16 + 14820239*t^17 + 1876473*t^18 - 38922813*t^19 + O(t^20)

There might be typo in your expression (some parentheses are open but not closed).