分割数の閉じた公式 - 書き下しバージョン

$\displaystyle \mathrm{P}(n) = \sum_{m=1}^n n^{m-n} \sum_{i_1=1}^{n^n} \frac{\prod_{i_2=1}^n \left(\sum_{j_1=1}^{m}\prod_{j_2=1}^{\sqrt{\left(1+\left(\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)-n\left(\sum_{j_5=1}^{\left(\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)} \prod_{j_6=1}^{\left(j_3 n-\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)}0\right)-i_2\right)^2}}0\right)!}{\left(\sum_{i_2=1}^n \sum_{j_1=1}^{m}\prod_{j_2=1}^{\sqrt{\left(1+\left(\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)-n\left(\sum_{j_5=1}^{\left(\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)} \prod_{j_6=1}^{\left(j_3 n-\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)}0\right)-i_2\right)^2}}0\right)!} \prod_{i_3=1}^{\sqrt{\left(\sum_{i_2=1}^n i_2 \cdot \sum_{j_1=1}^{m}\prod_{j_2=1}^{\sqrt{\left(1+\left(\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)-n\left(\sum_{j_5=1}^{\left(\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)} \prod_{j_6=1}^{\left(j_3 n-\sum_{j_3=1}^{i_1} \prod_{j_4=1}^{j_3 n^{j_1-1}-i_1}0\right)}0\right)-i_2\right)^2}}0-n\right)^2}}0$

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http://d.hatena.ne.jp/cgi-bin/mimetex.cgi?\displaystyle~\mathrm{P}(n)~=~\sum_{m=1}^n~n^{m-n}~\sum_{i_1=1}^{n^n}~\frac{\prod_{i_2=1}^n~\left(\sum_{j_1=1}^{m}\prod_{j_2=1}^{\sqrt{\left(1+\left(\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)-n\left(\sum_{j_5=1}^{\left(\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)}~\prod_{j_6=1}^{\left(j_3~n-\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)}0\right)-i_2\right)^2}}0\right)!}{\left(\sum_{i_2=1}^n~\sum_{j_1=1}^{m}\prod_{j_2=1}^{\sqrt{\left(1+\left(\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)-n\left(\sum_{j_5=1}^{\left(\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)}~\prod_{j_6=1}^{\left(j_3~n-\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)}0\right)-i_2\right)^2}}0\right)!}~\prod_{i_3=1}^{\sqrt{\left(\sum_{i_2=1}^n~i_2~\cdot~\sum_{j_1=1}^{m}\prod_{j_2=1}^{\sqrt{\left(1+\left(\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)-n\left(\sum_{j_5=1}^{\left(\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)}~\prod_{j_6=1}^{\left(j_3~n-\sum_{j_3=1}^{i_1}~\prod_{j_4=1}^{j_3~n^{j_1-1}-i_1}0\right)}0\right)-i_2\right)^2}}0-n\right)^2}}0