五边形数定理

<p>关于欧拉函数的展开式 <img src="https://www.showdoc.com.cn/server/api/attachment/visitfile/sign/3dd377c0708fbfa56e372ba0eca840cd" alt="" /></p> <p>与分割函数$$p(x)$$（x分成多少个自然数之和）关系 <img src="https://www.showdoc.com.cn/server/api/attachment/visitfile/sign/65b7cf7912235367c68f906e4032be92" alt="" /> 有 <img src="https://www.showdoc.com.cn/server/api/attachment/visitfile/sign/3e66a5de9e507951f6e4501d7d99e1e3" alt="" /> 可得 <img src="https://www.showdoc.com.cn/server/api/attachment/visitfile/sign/de6ce4c7190f6e1892a3658cbb826400" alt="" /></p> <pre><code>$$\prod\limits_{n=1}^{\infty}(1-x^n)=\sum\limits_{k=1}^{\infty}(-1)^kx^{\frac{k(k\pm 1)}{2}}$$ $$=1-x-x^2+x^5+x^7-x^{12}-x^{15}+x^{22}+x^{26}\dots$$</code></pre> <pre><code>$$\frac{1}{\phi(x)}=\sum\limits_{k=0}^{\infty}p(x)x^k$$ $$(1-x-x^2+x^5+x^7-x^{12}-x^{15}+x^{22}\dots)(1+p(1)x+p(2)x^2\dots)=1$$ $$p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+...$$</code></pre>