= Bernoulli number
{c}
{title2=$B_n$}
{wiki}
The <Bernoulli numbers> are a <sequence> of <rational numbers> that turns <integer power sums> into exact <polynomials>. The same numbers describe corrections when a discrete <summation> is compared with an <integral>. For example, the <sum of squares> $1^2+\cdots+N^2$ is $N^3/3+N^2/2+N/6$: the leading term is an <integral>, while the last two terms repair the difference between area and sampled values.
We use \b[$B_1=-\tfrac12$ throughout]. The defining <exponential generating function> is
$$
\frac{t}{e^t-1}=\sum_{n=0}^{\infty}B_n\frac{t^n}{n!}
=1-\frac t2+\frac{t^2}{12}-\frac{t^4}{720}+\frac{t^6}{30240}-\cdots.
$$
Here $e^t$ is the <exponential function> and $n!$ is a <factorial>; in particular, the <coefficient> of $t^2$ is $B_2/2!$, rather than $B_2$. At $t=0$ the quotient has limiting value one. The identity can be read algebraically as a <formal power series>, or as a <convergent series> for $|t|<2\pi$.<NIST Digital Library of Mathematical Functions>[Karl Dilcher, Chapter 24, §24.2(i), equation 24.2.1: defining generating function and convergence domain]{ref}
To compute the numbers, multiply by $e^t-1$ and compare <coefficients>. The resulting <recurrence relation>, with <binomial coefficients>, is
$$
B_0=1,\qquad \sum_{k=0}^{m}\binom{m+1}{k}B_k=0\quad(m\geq1),\qquad
B_m=-\frac{1}{m+1}\sum_{k=0}^{m-1}\binom{m+1}{k}B_k.
$$
Thus $1+2B_1=0$, $1+3B_1+3B_2=0$, and $1+4B_1+6B_2+4B_3=0$ give $B_1=-1/2$, $B_2=1/6$, and $B_3=0$. Continuing gives
$$
\begin{array}{c|rrrrrrrrrrr}
n&0&1&2&3&4&5&6&7&8&9&10\\\hline
B_n&1&-\tfrac12&\tfrac16&0&-\tfrac1{30}&0&\tfrac1{42}&0&-\tfrac1{30}&0&\tfrac5{66}
\end{array}
$$
The <recurrence relation> agrees with equation 24.5.3 after replacing its index $n$ by $m+1$.<NIST Digital Library of Mathematical Functions>[Karl Dilcher, Chapter 24, §24.5(i), equation 24.5.3; first values also in Table 24.2.1]{ref}
Why do so many values vanish? Put $F(t)=t/(e^t-1)$. Direct substitution gives $F(-t)=F(t)+t$, so $F(t)+t/2$ is an <even function>. Its odd <power series> <coefficients> vanish: $B_{2r+1}=0$ for every <positive integer> $r$. The exception $B_1$ matters. The alternative convention obtained from $t/(1-e^{-t})=F(t)+t$ has $B_1^{+}=+1/2$ and exactly the same other values. A formula copied between conventions must also retain its summation endpoints.
Historically, <integer power sums> were studied before the modern <Bernoulli number> notation. A historical study examines formulas published in 1631 and distinguishes those separate formulas from encoding all exponents using one <sequence> of constants. This is a useful motivation for the <generating function>, without assigning all discovery to a single moment. <Johann Faulhaber and sums of powers by Donald E. Knuth>[Donald E. Knuth, Johann Faulhaber and sums of powers, 1992 preprint, pp. 2 and 14: the 1631 power-sum formulas and the uniform Bernoulli-number formula]{ref}
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