Pensieve header: Verifying the Quesne formula.

c[k_] := ((((1-q))^k)/(k(1-(q^k))));
Table[c[k], {k,1,10}]

a[n_] := a[n]=Together@SeriesCoefficient[Exp[(∑_(k=1)^n)c[k](x^k)], {x,0,n}];

Table[Echo[n->a[n]], {n,0,10}];

Table[FunctionExpand@QFactorial[n,q], {n,0,6}]

Table[a[n]FunctionExpand@QFactorial[n,q], {n,0,10}]