Introductio in analysin infinitorum

NUMERORUM. 257 minorum in Serie $\alpha, \beta, \gamma, \delta, \epsilon, \text{&c.}$, contentorum produci queat. Hoc igitur pacto quaestio priori, quam ante sumus contemplati, similis resolvetur. CAP. XVI. 304. Accommodemus haec ad casum inprimis notatu dignum, sitque proposita haec expressio $\frac{1}{(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)\text{&c.}}$ quae per divisionem evoluta dabit $1+z(x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+\text{&c.})$ $+z^2(x^2+x^3+2x^4+2x^5+3x^6+3x^7+4x^8+4x^9+5x^{10}+\text{&c.})$ $+z^3(x^3+x^4+2x^5+3x^6+4x^7+5x^8+7x^9+8x^{10}+10x^{11}+\text{&c.})$ $+z^4(x^4+x^5+2x^6+3x^7+5x^8+6x^9+9x^{10}+11x^{11}+15x^{12}+\text{&c.})$ $+z^5(x^5+x^6+2x^7+3x^8+5x^9+7x^{10}+10x^{11}+13x^{12}+18x^{13}+\text{&c.})$ $+z^6(x^6+x^7+2x^8+3x^9+5x^{10}+7x^{11}+11x^{12}+14x^{13}+20x^{14}+\text{&c.})$ $+z^7(x^7+x^8+2x^9+3x^{10}+5x^{11}+7x^{12}+11x^{13}+15x^{14}+21x^{15}+\text{&c.})$ $+z^8(x^8+x^9+2x^{10}+3x^{11}+5x^{12}+7x^{13}+11x^{14}+15x^{15}+22x^{16}+\text{&c.})$ &c., Ex his ergo Seriebus statim definire licet quot variis modis propositus numerus per additionem ex dato terminorum hujus Seriei $1, 2, 3, 4, 5, 6, 7, \text{&c.}$, numero produci queat. Sic, si quaeratur quot variis modis numerus $13$ oriri possit per additionem quinque numerorum integrorum, spectari debebit terminus $x^{13}z^5$, cujus coefficiens $18$ indicat numerum propositum $13$ ex quinque numerorum additione octodecim modis oriri posse. 305. Si ponatur $z=1$, atque similes Potestates ipsius $x$ conjunctim exprimantur, haec expressio $\frac{1}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)(1-x^6)\text{&c.}}$ evolvetur in hanc Seriem Euleri Introduct. in Anal. infin. parv. Kk $1+$