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  "body": "\n## Definition\n\nThe Bernoulli numbers $B_0, B_1, B_2, \\dots$ are the rational numbers given by the power series expansion\n$$\\frac{t}{e^t - 1} = \\sum_{n=0}^{\\infty} B_n \\frac{t^n}{n!}.$$\nEquivalently, $B_0 = 1$ and $\\sum_{k=0}^{n} \\binom{n+1}{k} B_k = 0$ for each $n \\ge 1$.\n\n## Remarks\n\nThe first values are $B_0 = 1$, $B_1 = -\\frac{1}{2}$, $B_2 = \\frac{1}{6}$, $B_4 = -\\frac{1}{30}$ and $B_6 = \\frac{1}{42}$. Also $B_n = 0$ for each odd $n \\ge 3$.\nSome authors use the convention $B_1 = +\\frac{1}{2}$, from the series $\\frac{t e^t}{e^t - 1}$.\nThey appear in the formulas for the sums of powers $1^k + 2^k + \\dots + n^k$, and in the values $\\zeta(2n)$ of the Riemann zeta function.\nThey are named after Jacob Bernoulli. Seki Takakazu tabulated them in his Katsuyō Sanpō of 1712.\n\n## Explanations\n\n### L1 human\n\nThe Bernoulli numbers are a sequence of fractions that appear in many formulas, for example in the formula for $1^k + 2^k + \\dots + n^k$.\nYou can compute them one after another from a simple recurrence.\n",
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