"""Failure mode: the other Bernoulli convention.

This is a routine a language model can produce when the prompt does not name
a convention: one that is correct under the *second* Bernoulli convention
(``B1 = +1/2``) when the specification asked for the first (``B1 = -1/2``).

Nothing here is mathematically wrong.  It is wrong *for this specification*, and
that distinction is the entire point.  A single unstated convention makes two
correct programs disagree, and no amount of re-reading the code will reveal it if
you do not know the convention exists.

Expected checker diagnosis: ``b1-sign``, with a first divergence at ``n=1``.
"""

from fractions import Fraction
from math import comb


def bernoulli(n: int) -> Fraction:
    """Return B_n using the second convention (B1 = +1/2)."""
    if n < 0:
        raise ValueError("n must be >= 0")

    values = [Fraction(0)] * (n + 1)
    for m in range(n + 1):
        values[m] = Fraction(1)
        for k in range(m):
            values[m] -= Fraction(comb(m, k) * values[k], m - k + 1)
    return values[n]