On the power-series expansion of a rational function

Introduction. The problem of determining the formula for ${\displaystyle P_S\left(n\right)}$, the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, ${\displaystyle h_{s₁},...,h_{s_k}}$, of the equation h_{s₁} s₁ + ... + h_{s_k} s_k = n, was solved in the nineteenth century (see Sylvester [4] and Glaisher [3] for detailed accounts). The solution is the coefficient of${\displaystyle xⁿ\in {[(1-x^{s₁})...(1-x^{s_k})]}^{-1},\exp ressionsf\text{or}whichtheyderived.Wright\left[5\right]\in dicateda\sim p\le rmethodbywhich\to f\in dpartofthesolution(at\le \ast \in thecase}$s_i=i${\displaystyle ).Thecurrentpapergivesa\sim p\le methodbywhichthepower-series\exp ansionofarationalfunctionmaybederived.Lemma1iswellknown\text{and}givesthe\ge \ne ralf\text{or}mofthesolution.Lemma2isalsowellknown.See,f\text{or}exa\mp \le ,Andrews\left[1\right],Exa\mp \le 2,p.98.Lemma3showshowtherecurrencerelationofLemma2becomesofboundeddegree\in certa\in cases.Therecurrencerelationisthensolved,\text{and}thesolutionisextendedomthesecerta\in cases\to allcases.Wethenapplytheres\underline{t}\to \in vestigatethegrowthofthed\iff erence}$P_S(n) - P_T(n)${\displaystyle ,whereS\text{and}Taref\in itesets,\text{and}\in partic\underline{a}rwhenthisd\iff erenceisbounded.Thed\iff erences}$P_S^{(0)}(n) - P_T^{(0)}(n)${\displaystyle \text{and}}$P_S^{(1)}(n) - P_T^{(1)}(n)${\displaystyle arealsoconsred,where}$P_S^{(0)}${\displaystyle (resp.}$P_S^{(1)}!$!) denotes the number of partitions of n into elements of S with an even (resp. odd) number of parts.