Introductio in analysin infinitorum

270 DE PARTITIONE LIB. I. hinc scala relationis habeatur. Quod si autem Factores denominatoris continuo in se multiplicentur, prodibit $1-x-x^2+x^5+x^7-x^{12}-x^{15}+x^{22}+x^{26}-x^{35}-x^{40}+x^{51}+\&c.$, quae Series si attentius consideretur, aliae Potestates ipsius $x$ adesse non deprehenduntur, nisi quarum Exponentes contineantur in hac formula $\frac{3nn \pm n}{2}$; atque, si $n$ sit numerus impar, Potestates erunt negativae; affirmativae autem si $n$ fuerit numerus par. 324. Cum igitur scala relationis sit $+1, +1, 0, 0, -1, 0, -1, 0, 0, 0, 0, +1, 0, 0, +1, 0, 0, \&c.$, Series recurrens ex evolutione fractionis $\frac{1}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)(1-x^6)(1-x^7)\&c.}$ oriunda erit haec $1+x+2x^2+3x^3+5x^4+7x^5+11x^6+15x^7+22x^8+30x^9+42x^{10}+56x^{11}+77x^{12}+101x^{13}+135x^{14}+176x^{15}+231x^{16}+297x^{17}+385x^{18}+490x^{19}+627x^{20}+792x^{21}+1002x^{22}+1250x^{23}+1570x^{24}+\&c.$ In hac ergo Serie coëfficiens quisque indicat, quot variis modis Exponens ipsius $x$ per additionem ex numeris integris oriri queat. Sic numerus 7 quindecim modis per additionem oriri potest. $7=7$ | $7=4+2+1$ | $7=3+1+1+1+1$ $7=6+1$ | $7=4+1+1+1$ | $7=2+2+2+1$ $7=5+2$ | $7=3+3+1$ | $7=2+2+1+1+1$ $7=5+1+1$ | $7=3+2+2$ | $7=2+1+1+1+1+1$ $7=4+3$ | $7=3+2+1+1$ | $7=1+1+1+1+1+1+1$ 325. Quod