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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 numbers $x$ with $\\abs{x} \u003c 1$, where both sides converge absolutely.\n\n## Remarks\n\nThe exponents $k(3k - 1)/2$ for $k = 1, -1, 2, -2, 3, -3, \\ldots$ are $1, 2, 5, 7, 12, 15, \\ldots$, the generalised pentagonal numbers.\nFranklin's bijective proof reads the identity as a statement on [[integer partitions|0JV30XN73Z]] of $m$ into distinct parts.\nThe partitions with an even number of parts and those with an odd number of parts are equinumerous, except when $m = k(3k \\pm 1)/2$.\nIn that case the difference, even minus odd, is $(-1)^k$.\nFranklin's involution moves the smallest part onto the largest parts, or back, and it changes the number of parts by one.\n\nExample: for $m = 5$, the partitions into distinct parts are $5$, $4 + 1$ and $3 + 2$.\nOne has an odd number of parts and two have an even number, so the coefficient of $x^5$ is $2 - 1 = 1$, as in the series.\n\nMultiplying by the [[generating function for partitions|0578PA53VA]] $\\prod_n (1 - x^n)^{-1}$ gives a recurrence for the [[partition function|0CV69WDRXW]]:\n$$p(m) = \\sum_{k \\ne 0} (-1)^{k+1}\\, p\\big(m - k(3k - 1)/2\\big),$$\nwith $p(0) = 1$ and $p(j) = 0$ for $j \u003c 0$.\nFor example, $p(5) = p(4) + p(3) - p(0) = 5 + 3 - 1 = 7$.\nThe theorem is a special case of the Jacobi triple product.\n\n## Explanations\n\n### L1 human\n\nMultiply out $(1 - x)(1 - x^2)(1 - x^3)\\cdots$ without end.\nOne expects a messy series, but almost everything cancels.\nOnly the powers $x^1, x^2, x^5, x^7, x^{12}, x^{15}, \\ldots$ survive, with signs in the pattern $-, -, +, +, -, -, \\ldots$, and every coefficient is $0$ or $\\pm 1$.\nThe surviving exponents are the pentagonal numbers and their close relatives.\n\n### L2 human\n\nThe product $\\prod (1 - x^n)$ is the reciprocal of the partition generating function, so the theorem controls the partition numbers $p(m)$.\nThe recurrence above computes $p(m)$ with about $\\sqrt{m}$ terms, which was the standard method for tables of $p(m)$.\nThe bijective proof by Franklin is a model of a sign-reversing involution.\nThe identity is the simplest of a family of product-sum identities that includes the Jacobi triple product and the [[Rogers–Ramanujan identities|M24H4GJPPW]].\n",
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