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  "body": "\n## Statement\n\nIn the ring of [[formal power series|033WGN00BF]] $\\Z[\\![x]\\!]$,\n$$\\prod_{n=1}^{\\infty} (1 - x^n) = \\sum_{k=-\\infty}^{\\infty} (-1)^k x^{k(3k-1)/2} = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + \\cdots.$$\nThe same identity holds for complex $x$ with $\\abs{x} \u003c 1$, both sides converging absolutely.\n\n## Remarks\n\nThe exponents $k(3k-1)/2$ for $k = 0, \\pm 1, \\pm 2, \\ldots$ are the generalized pentagonal numbers $0, 1, 2, 5, 7, 12, 15, 22, 26, \\ldots$.\nLet $E(m)$ and $O(m)$ be the numbers of partitions of $m$ into an even and an odd number of distinct parts.\nThe coefficient of $x^m$ on the left is $E(m) - O(m)$.\nFranklin's bijection pairs these two kinds of partitions, and the only unpaired partitions occur when $m$ is a generalized pentagonal number, with sign $(-1)^k$.\nExample: for $m = 3$, the distinct-part partitions are $3$ (one part) and $2 + 1$ (two parts), so the coefficient is $1 - 1 = 0$; indeed $3$ is not a generalized pentagonal number.\nThe [[generating function for partitions|0578PA53VA]] is $\\prod_n (1 - x^n)^{-1} = \\sum_m p(m) x^m$.\nMultiplying the theorem by it gives the recurrence\n$$p(m) = p(m-1) + p(m-2) - p(m-5) - p(m-7) + p(m-12) + p(m-15) - \\cdots,$$\nwith $p(0) = 1$ and $p(j) = 0$ for $j \u003c 0$.\n\n## Explanations\n\n### L1 human\n\nMultiply out $(1 - x)(1 - x^2)(1 - x^3)\\cdots$.\nOne expects a mess, but almost everything cancels: only the powers $0, 1, 2, 5, 7, 12, 15, \\ldots$ survive, each with coefficient $+1$ or $-1$.\nThese exponents are the pentagonal numbers, which count dots in nested pentagons, together with a second family of the same shape.\n\n### L2 human\n\nEuler found the identity, and it gives a fast recurrence for the [[partition function|001CX45SWQ]] $p(m)$.\nIt is a special case of the Jacobi triple product identity.\nThe left side is, up to the factor $x^{1/24}$, the [[Dedekind eta function|MX3CVYMNDV]], so the theorem is also a statement about a modular form.\n",
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