Partition function (number theory): Difference between revisions

Revision as of 22:48, 7 August 2024

Revision as of 23:39, 12 December 2024

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<math display="block">\begin{align}

\sum_{n=0}^\infty p(n)x^n &= \prod_{k=1}^\infty \left(\frac {1}{1-x^k} \right)\\

&=\left(1+x+x^2+x^3+\cdots\right) \left(1+x^2+x^4+x^6+\cdots\right)
\left(1+x^3+x^6+x^9+\cdots\right) \cdots \\

&=\frac{1}{1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26} - \cdots}\\

&=1 \Big/ \sum_{k=-\infty}^{\infty} (-1)^k x^{k(3k-1)/2}.