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Bernoulli numbers
The Bernoulli numbers rational numbers arising from the Bernoulli polynomials, are (some authors use )
Numerators of Bernoulli numbers are listed in A027641, denominators in A027642.
The even indexed Bernoulli numbers are (some authors write for )
Numerators of even indexed Bernoulli numbers are listed in A000367, denominators in A002445.
The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712 in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713.
Ada Lovelace's note G on the analytical engine from 1842 describes an algorithm for generating Bernoulli numbers with Charles Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first computer program.
Contents
Sum of powers
Closed forms of the sum of powers for fixed values of
are always polynomials in of degree , called Bernoulli polynomials. Note that for all since in this case the sum is the empty sum.
The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula
where (instead of ).
Some authors state Bernoulli's formula in a different way
where .
Let . Taking to be 0 and gives the natural numbers {0, 1, 2, 3, ...} (A001477).
Taking to be 1 and (instead of ) gives the triangular numbers {0, 1, 3, 6, ...} (A000217).
Taking to be 2 and gives the square pyramidal numbers {0, 1, 5, 14, ...} (A000330).
Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sum of powers.
Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005).
Sum of powers triangle
= 0 | 1 | ||||||||||
1 | 1 | 1 | |||||||||
2 | 1 | 3 | 2 | ||||||||
3 | 1 | 7 | 12 | 6 | |||||||
4 | 1 | 15 | 50 | 60 | 24 | ||||||
5 | 1 | 31 | 180 | 390 | 360 | 120 | |||||
6 | 1 | 63 | 602 | 2100 | 3360 | 2520 | 720 | ||||
7 | 1 | 127 | 1932 | 10206 | 25200 | 31920 | 20160 | 5040 | |||
8 | 1 | 255 | 6050 | 46620 | 166824 | 317520 | 332640 | 181440 | 40320 | ||
9 | 1 | 511 | ? | ? | ? | ? | ? | ? | ? | 362880 | |
10 | 1 | 1023 | ? | ? | ? | ? | ? | ? | ? | ? | 3628800 |
Examples:
- C(3, 1) * a(5, 0) + C(3, 2) * a(5, 1) + C(3, 3) * a(5, 2) = 3 * 1 + 3 * 31 + 1 * 180 = 276
- S_5(3) = 1^5 + 2^5 + 3^5 = 276
- C(4, 1) * a(5, 0) + C(4, 2) * a(5, 1) + C(4, 3) * a(5, 2) + C(4, 4) * a(5, 3) = 4 * 1 + 6 * 31 + 4 * 180 + 1 * 390 = 1300
- S_5(4) = 1^5 + 2^5 + 3^5 + 4^5 = 1300
A028246 Triangular array of numbers a(n,k) = (1/k) * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * i^n; n >= 1, 1 <= k <= n.
- {1, 1, 1, 1, 3, 2, 1, 7, 12, 6, 1, 15, 50, 60, 24, 1, 31, 180, 390, 360, 120, 1, 63, 602, 2100, 3360, 2520, 720, 1, 127, 1932, 10206, 25200, 31920, 20160, 5040, ...}
Generating function
Exponential generating function
The exponential generating function for the Bernoulli numbers is
Asymptotic approximation
For even the Bernoulli numbers can be approximated by
This formula (Peter Luschny, 2007) is based on the well known connection of the Bernoulli numbers with the Riemann zeta function and on an approximation of the factorial function given by Gergő Nemes[1] in 2007 (A181855/A181856). For example this approximation gives
which is off only by three units in the least significant digit displayed.
This formula is an improvement over a standard asymptotic formula for the even Bernoulli numbers (see DLMF/NIST[2])
Sequences
A027641 Numerators of Bernoulli numbers .
- {1, –1, 1, 0, –1, 0, 1, 0, –1, 0, 5, 0, –691, 0, 7, 0, –3617, 0, 43867, 0, –174611, 0, 854513, 0, –236364091, 0, 8553103, 0, –23749461029, 0, 8615841276005, 0, –7709321041217, ...}
A027642 Denominators of Bernoulli numbers .
- {1, 2, 6, 1, 30, 1, 42, 1, 30, 1, 66, 1, 2730, 1, 6, 1, 510, 1, 798, 1, 330, 1, 138, 1, 2730, 1, 6, 1, 870, 1, 14322, 1, 510, 1, 6, 1, 1919190, 1, 6, 1, 13530, 1, 1806, 1, 690, ...}
A000367 Numerators of even indexed Bernoulli numbers .
- {1, 1, –1, 1, –1, 5, –691, 7, –3617, 43867, –174611, 854513, –236364091, 8553103, –23749461029, 8615841276005, –7709321041217, 2577687858367, –26315271553053477373, ...}
A002445 Denominators of even indexed Bernoulli numbers .
- {1, 6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, 1806, 690, 282, 46410, 66, 1590, 798, 870, 354, 56786730, 6, 510, 64722, 30, ...}
A?????? Numerators of odd indexed Bernoulli numbers .
- {–1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}
A054977 Denominators of odd indexed Bernoulli numbers .
- {2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
Notes
External links
- Bernd C. Kellner, The Bernoulli Number Page.
- Guo, Victor J. W.; Zeng, Jiang (2005), “A q-Analogue of Faulhaber's Formula for Sums of Powers”, The Electronic Journal of Combinatorics 11 (2): 1441, Bibcode: 2005math......1441G.