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Bernoulli B₆
1/42
Sixth Bernoulli number. Modern convention B₆ = 1/42.
Jakob Bernoulli was adding up 1² + 2² + 3² + … and looking for a shortcut. The numbers that fall out of that work are named for him. The sixth one is exactly 1/42. Not cake. Not Catalan. A fraction analysis keeps handing back.
What it supports
An exact rational in a named sequence. Modern convention (B₁ = −1/2): B₀=1, B₂=1/6, B₄=−1/30, B₆=1/42. They appear in power sums, Euler-Maclaurin summation, and even values of the Riemann zeta function. 42 is an early record denominator in the sequence (MathWorld). Independent of Catalan C₅, which counts noncrossing ways.
What it does not
Not Catalan. Not the cake number. Not Adams. Odd indices after B₁ are zero; that is the sequence, not a miss.
Calculation
B₆ = 1/42. Generating function x/(e^x − 1) = Σ Bₙ xⁿ / n!.
Modern even-index Bernoulli numbers. Ars Conjectandi, 1713.
ExactnessExact
Form42
Unitrational
Strengthhard
When1713