Going further, we can get recurrence formulas for \( \mathrm{smag}_{n,j} \). We have
\( N_n = \sum_{j=n-1}^{\frac{n(n-1)}{2}} \mathrm{smag}_{n,j} u^j \).
\( \begin{align}
\sum_{j=n-1}^{\frac{n(n-1)}{2}} \mathrm{smag}_{n,j} u^j &= \sum_{m=1}^{n-1} (-1)^{n-m} \left(\prod_{j=m}^{n-2} (1 - u^j)\right) u^{n-1} \left{{n \atop m}\right} N_m\\
&= \sum_{m=1}^{n-1} (-1)^{n-m} u^{n-1} \left{{n \atop m}\right} \left(\prod_{j=m}^{n-2} (1 - u^j)\right) \left(\sum_{j=m-1}^{\frac{m(m-1)}{2}} \mathrm{smag}_{m,j} u^j\right)\\
&= \sum_{m=1}^{n-1} (-1)^{n-m} u^{n-1} \left{{n \atop m}\right} \left(\sum_{j=0}^{\frac{(n-1)(n-2) - m(m-1)}{2}} R_{m,n-2,j} u^j\right) \left(\sum_{j=m-1}^{\frac{m(m-1)}{2}} \mathrm{smag}_{m,j} u^j\right)\\
&= \sum_{m=1}^{n-1} (-1)^{n-m} \left{{n \atop m}\right} \left(\sum_{j=n-1}^{\frac{n(n-1) - m(m-1)}{2}} R_{m,n-2,j-n+1} u^j\right) \left(\sum_{j=m-1}^{\frac{m(m-1)}{2}} \mathrm{smag}_{m,j} u^j\right)\\
\end{align} \).
Now consider all occurrences of \( u^j \) for a given j. That will be the recurrence for generating the \( \mathrm{smag} \). Let's see where that goes. We multiply the two sums together to get
\( \left(\sum_{j=n-1}^{\frac{n(n-1) - m(m-1)}{2}} R_{m,n-2,j-n+1} u^j\right) \left(\sum_{j=m-1}^{\frac{m(m-1)}{2}} \mathrm{smag}_{m,j} u^j\right) = \sum_{j=n+m-2}^{\frac{n(n-1)}{2}} \left(\sum_{k,l:\ k+l = j \\ n-1 \le k \le \frac{n(n-1) - m(m-1)}{2} \\ m-1 \le l \le \frac{m(m-1)}{2}} R_{m,n-2,k-n+1} \mathrm{smag}_{m,l}\right) u^j \).
Then we have
\( \sum_{j=n-1}^{\frac{n(n-1)}{2}} \mathrm{smag}_{n,j} u^j = \sum_{m=1}^{n-1} (-1)^{n-m} \left{{n \atop m}\right} \sum_{j=n+m-2}^{\frac{n(n-1)}{2}} \left(\sum_{k,l:\ k+l = j \\ n-1 \le k \le \frac{n(n-1) - m(m-1)}{2} \\ m-1 \le l \le \frac{m(m-1)}{2}} R_{m,n-2,k-n+1} \mathrm{smag}_{m,l}\right) u^j
\).
To get the coefficient for a given j, we note that the sum over j will only include that j when \( n + m - 2 \le j \), which means \( m \le j-n+2 \). So we get
\( \mathrm{smag}_{n,j} = \sum_{m=1}^{j-n+2} (-1)^{n-m} \left{{n \atop m}\right} \left(\sum_{k,l:\ k+l = j \\ n-1 \le k \le \frac{n(n-1) - m(m-1)}{2} \\ m-1 \le l \le \frac{m(m-1)}{2}} R_{m,n-2,k-n+1} \mathrm{smag}_{m,l}\right) \)
as the atrocious recurrence for the magic numbers. What to do with that?
(P.S. \( R_{m,n,j} \) when \( n < m \) should be 1)
\( N_n = \sum_{j=n-1}^{\frac{n(n-1)}{2}} \mathrm{smag}_{n,j} u^j \).
\( \begin{align}
\sum_{j=n-1}^{\frac{n(n-1)}{2}} \mathrm{smag}_{n,j} u^j &= \sum_{m=1}^{n-1} (-1)^{n-m} \left(\prod_{j=m}^{n-2} (1 - u^j)\right) u^{n-1} \left{{n \atop m}\right} N_m\\
&= \sum_{m=1}^{n-1} (-1)^{n-m} u^{n-1} \left{{n \atop m}\right} \left(\prod_{j=m}^{n-2} (1 - u^j)\right) \left(\sum_{j=m-1}^{\frac{m(m-1)}{2}} \mathrm{smag}_{m,j} u^j\right)\\
&= \sum_{m=1}^{n-1} (-1)^{n-m} u^{n-1} \left{{n \atop m}\right} \left(\sum_{j=0}^{\frac{(n-1)(n-2) - m(m-1)}{2}} R_{m,n-2,j} u^j\right) \left(\sum_{j=m-1}^{\frac{m(m-1)}{2}} \mathrm{smag}_{m,j} u^j\right)\\
&= \sum_{m=1}^{n-1} (-1)^{n-m} \left{{n \atop m}\right} \left(\sum_{j=n-1}^{\frac{n(n-1) - m(m-1)}{2}} R_{m,n-2,j-n+1} u^j\right) \left(\sum_{j=m-1}^{\frac{m(m-1)}{2}} \mathrm{smag}_{m,j} u^j\right)\\
\end{align} \).
Now consider all occurrences of \( u^j \) for a given j. That will be the recurrence for generating the \( \mathrm{smag} \). Let's see where that goes. We multiply the two sums together to get
\( \left(\sum_{j=n-1}^{\frac{n(n-1) - m(m-1)}{2}} R_{m,n-2,j-n+1} u^j\right) \left(\sum_{j=m-1}^{\frac{m(m-1)}{2}} \mathrm{smag}_{m,j} u^j\right) = \sum_{j=n+m-2}^{\frac{n(n-1)}{2}} \left(\sum_{k,l:\ k+l = j \\ n-1 \le k \le \frac{n(n-1) - m(m-1)}{2} \\ m-1 \le l \le \frac{m(m-1)}{2}} R_{m,n-2,k-n+1} \mathrm{smag}_{m,l}\right) u^j \).
Then we have
\( \sum_{j=n-1}^{\frac{n(n-1)}{2}} \mathrm{smag}_{n,j} u^j = \sum_{m=1}^{n-1} (-1)^{n-m} \left{{n \atop m}\right} \sum_{j=n+m-2}^{\frac{n(n-1)}{2}} \left(\sum_{k,l:\ k+l = j \\ n-1 \le k \le \frac{n(n-1) - m(m-1)}{2} \\ m-1 \le l \le \frac{m(m-1)}{2}} R_{m,n-2,k-n+1} \mathrm{smag}_{m,l}\right) u^j
\).
To get the coefficient for a given j, we note that the sum over j will only include that j when \( n + m - 2 \le j \), which means \( m \le j-n+2 \). So we get
\( \mathrm{smag}_{n,j} = \sum_{m=1}^{j-n+2} (-1)^{n-m} \left{{n \atop m}\right} \left(\sum_{k,l:\ k+l = j \\ n-1 \le k \le \frac{n(n-1) - m(m-1)}{2} \\ m-1 \le l \le \frac{m(m-1)}{2}} R_{m,n-2,k-n+1} \mathrm{smag}_{m,l}\right) \)
as the atrocious recurrence for the magic numbers. What to do with that?
(P.S. \( R_{m,n,j} \) when \( n < m \) should be 1)
