An identity for the divisor generating function arising from sorting theory

In the present paper we prove

$\text{∑}\text{n=1}\text{∞}\text{nx}{}^{\mathrm{n}}\text{∏}\text{j=n+I}\text{∞}\text{(1?x}{}^{\mathrm{j}}\text{)}$

$\text{=}\text{∑}\text{n=1}\text{∞}\text{(?1)}{}^{\mathrm{n?1}}\text{x}{}^{\mathrm{<mtext>n(n+1)</mtext><mtext>2</mtext>}}\text{(1?x)(1?x}{}^2\text{)?(1?x}{}^{\mathrm{n?1}}\text{)(1?x}{}^{\mathrm{n}}\text{)}{}^2$

$\text{=}\text{∑}\text{n=1}\text{∞}\text{d(n)x}{}^{\mathrm{n}}$