closed form for product of series

Question

Is there a formula expressing the coefficients of the product $$\prod_{k=1}^n \left(\sum_{j=0}^\infty (j+1)x^{jk}\right),$$ where $n\ge 1$, in terms of sums of products of integer partitions?

Answer 1

Let $$ f_n(x)=\prod_{k=1}^n(1-x^k)^{-1} $$ It is well known that $$ f_n(x)=\sum_{j\ge 0} p_{\le n}(j) x^j $$ where $p_n(j)$ is the number of partitions of $j$ into parts whose sizes are all at most $n$. Equivalently, via conjugation, $p_n(j)$ is the number of partitions of $j$ into parts at most $n$ parts.

Your generating function is $$ \prod_{k=1}^n \left(\sum_{j=0}^\infty (j+1)x^{jk}\right)=\prod_{k=1}^n(1-x^k)^{-2}=[f_n(x)]^2 $$ Therefore, since the product of generating functions leads to a convolution of coefficient sequences, we conclude $$ \text{coefficient of $x^m$ in $[f_n(x)]^2$}=p_{\le n}(0)p_{\le n}(m)+p_{\le n}(1)p_{\le n}(m-1)+\dots+p_{\le n}(m)p_{\le n}(0) $$

Answer 2

Answering the question in title, using Pochhammer symbols $$f(n)=\prod_{k=1}^n \left(\sum_{j=0}^\infty (j+1)x^{jk}\right)=\frac{1}{\left((x;x)_n\right){}^2}$$

What is interestin if to look at the first terms of the expansions $$\left( \begin{array}{cc} 1 & 1+2 x+3 x^2+4 x^3+5 x^4+6 x^5+7 x^6+8 x^7+9 x^8+10 x^9+11 x^{10} \\ 2 & 1+2 x+5 x^2+8 x^3+14 x^4+20 x^5+30 x^6+40 x^7+55 x^8+70 x^9+91 x^{10}\\ 3 & 1+2 x+5 x^2+10 x^3+18 x^4+30 x^5+49 x^6+74 x^7+110 x^8+158 x^9+221 x^{10} \\ 4 & 1+2 x+5 x^2+10 x^3+20 x^4+34 x^5+59 x^6+94 x^7+149 x^8+224 x^9+334 x^{10} \\ 5 & 1+2 x+5 x^2+10 x^3+20 x^4+36 x^5+63 x^6+104 x^7+169 x^8+264 x^9+405 x^{10} \\ 6 & 1+2 x+5 x^2+10 x^3+20 x^4+36 x^5+65 x^6+108 x^7+179 x^8+284 x^9+445 x^{10} \\ 7 & 1+2 x+5 x^2+10 x^3+20 x^4+36 x^5+65 x^6+110 x^7+183 x^8+294 x^9+465 x^{10} \\ 8 & 1+2 x+5 x^2+10 x^3+20 x^4+36 x^5+65 x^6+110 x^7+185 x^8+298 x^9+475 x^{10}\\ 9 & 1+2 x+5 x^2+10 x^3+20 x^4+36 x^5+65 x^6+110 x^7+185 x^8+300 x^9+479 x^{10}\\ 10 & 1+2 x+5 x^2+10 x^3+20 x^4+36 x^5+65 x^6+110 x^7+185 x^8+300 x^9+481 x^{10}\\ 11 & 1+2 x+5 x^2+10 x^3+20 x^4+36 x^5+65 x^6+110 x^7+185 x^8+300 x^9+481 x^{10} \end{array} \right)$$ Finding a closed form of the coefficients of the expansion does not seem easy (at least to me).

Have a look at sequence $A000712$ is $OEIS$ which gives a table of the number of partitions of $n$ into parts of two kinds. No closed form given but a few asymptotics.