Introductio in analysin infinitorum

NUMERORUM. 255 $1+z(x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+\text{&c.})$ $+z^2(x^3+x^4+2x^5+2x^6+3x^7+3x^8+4x^9+4x^{10}+5x^{11}+\text{&c.})$ $+z^3(x^6+x^7+2x^8+2x^9+3x^{10}+4x^{11}+5x^{12}+7x^{13}+8x^{14}+10x^{15}+\text{&c.})$ $+z^4(x^{10}+x^{11}+2x^{12}+3x^{13}+5x^{14}+6x^{15}+9x^{16}+11x^{17}+15x^{18}+\text{&c.})$ $+z^5(x^{15}+x^{16}+2x^{17}+3x^{18}+5x^{19}+7x^{20}+10x^{21}+13x^{22}+18x^{23}+\text{&c.})$ $+z^6(x^{21}+x^{22}+2x^{23}+3x^{24}+5x^{25}+7x^{26}+11x^{27}+14x^{28}+20x^{29}+\text{&c.})$ $+z^7(x^{28}+x^{29}+2x^{30}+3x^{31}+5x^{32}+7x^{33}+11x^{34}+15x^{35}+21x^{36}+\text{&c.})$ $+z^8(x^{36}+x^{37}+2x^{38}+3x^{39}+5x^{40}+7x^{41}+11x^{42}+15x^{43}+22x^{44}+\text{&c.})$ &c. CAP. XVI. Ex his ergo Seriebus statim definire licet quot variis modis propositus numerus ex dato terminorum diversorum hujus Seriei $1, 2, 3, 4, 5, 6, 7, 8,$ &c., numero oriri queat. Sic, si quaeratur quot variis modis numerus $35$ possit esse summa septem terminorum diversorum Seriei $1, 2, 3, 4, 5, 6, 7,$ &c., quaeratur in Serie $z^7$ multiplicante Potestas $x^{35}$, ejusque coëfficiens $15$ indicabit numerum propositum $35$ quindecim variis modis esse summam septem terminorum Seriei $1, 2, 3, 4, 5, 6, 7, 8,$ &c. 301. Quod si autem ponatur $z=1$, & similes Potestates ipsius $x$ in unam summam conjiciantur, seu, quod eodem redit, si evolvatur haec expressio infinita $(1+x)(1+x^2)(1+x^3)(1+x^4)(1+x^5)(1+x^6)$ &c., quo facto orietur haec Series $1+x+x^2+2x^3+2x^4+3x^5+4x^6+5x^7+6x^8+\text{&c.},$ ubi quivis coëfficiens indicat, quot variis modis Exponens Potestatis ipsius $x$ conjunctae ex terminis diversis Seriei $1, 2, 3, 4, 5, 6, 7,$ &c., per additionem emergere possit. Sic apparet numerum $8$ sex modis per additionem diversorum numerorum produci, qui sunt $8 = 8$ $8 = 7+1$ $8 = 6+2$ $8 = 5+3$ $8 = 5+2+1$ $8 = 4+3+1$ ubi