Bernoulli numbers
Faulhaber formula coefficients
Bernoulli numbers are a sequence of rational numbers that appear throughout mathematics, particularly in number theory, calculus, and the study of sums of powers.
They emerge when calculating formulas for adding consecutive integers raised to a power: The sum of the first n cubes can be expressed using Bernoulli numbers.
Named after Swiss mathematician Jacob Bernoulli, they also appear in the Taylor series expansions of trigonometric functions and in deep connections to prime numbers through the Riemann zeta function.
Ordered succession of elements
- coefficientsNumerical multipliers in expressions
- Pascal's triangleTriangular array of binomial coefficients
- recurrence relationFormula where terms depend on earlier ones
- enumerationSequential listing of items
Adding to find total
- integrationCombining into a whole
- generating functionFormal series encoding sequences
- antiderivativeReverse of differentiation
- recurrence relationFormula where terms depend on earlier ones
Prolific 18th-century mathematician
- Ramanujan
- Ada Lovelace
- BabbageMechanical computer originator
- seventeenth centuryThird period of second millennium
sequence
Ordered succession of elements
- coefficientsNumerical multipliers in expressions
- Pascal's triangleTriangular array of binomial coefficients
- recurrence relationFormula where terms depend on earlier ones
summation
Adding to find total
- integrationCombining into a whole
- generating functionFormal series encoding sequences
- antiderivativeReverse of differentiation
Euler
Prolific 18th-century mathematician
- BabbageMechanical computer originator
- seventeenth centuryThird period of second millennium
Fermat's Last Theorem
No integer cubes sum to cubes
manifold
Many and various
- Bernoulli
- math historyEuclid, Pythagoras, Newton, Leibniz, Euler
- number sequencesFibonacci sequence, arithmetic sequence, integers, perfect squares, Pascal's triangle
- finite differences
- recursionrecursion, base case, self-reference, infinite recursion, recurrence relation
- seriesseries, sum, sequence, geometric series, arithmetic sequence
Bernoulli numbersStirling numbers
Counts set partitions or permutation cycles