Bernoulli numbers
edit on github · preview edits · show history · short url · generating function sequence special values
Numbers
$n$ 
$B_n$
0:
1
1:
-1/2
2:
1/6
3:
0
4:
-1/30
5:
0
6:
1/42
7:
0
8:
-1/30
9:
0
10:
5/66
11:
0
12:
-691/2730
13:
0
14:
7/6
15:
0
16:
-3617/510
17:
0
18:
43867/798
19:
0
20:
-174611/330
21:
0
22:
854513/138
23:
0
24:
-236364091/2730
25:
0
26:
8553103/6
27:
0
28:
-23749461029/870
29:
0
30:
8615841276005/14322
31:
0
32:
-7709321041217/510
33:
0
34:
2577687858367/6
35:
0
36:
-26315271553053477373/1919190
37:
0
38:
2929993913841559/6
39:
0
40:
-261082718496449122051/13530
41:
0
42:
1520097643918070802691/1806
43:
0
44:
-27833269579301024235023/690
45:
0
46:
596451111593912163277961/282
47:
0
48:
-5609403368997817686249127547/46410
49:
0
50:
495057205241079648212477525/66
51:
0
52:
-801165718135489957347924991853/1590
53:
0
54:
29149963634884862421418123812691/798
55:
0
56:
-2479392929313226753685415739663229/870
57:
0
58:
84483613348880041862046775994036021/354
59:
0
60:
-1215233140483755572040304994079820246041491/56786730
61:
0
62:
12300585434086858541953039857403386151/6
63:
0
64:
-106783830147866529886385444979142647942017/510
65:
0
66:
1472600022126335654051619428551932342241899101/64722
67:
0
68:
-78773130858718728141909149208474606244347001/30
69:
0
70:
1505381347333367003803076567377857208511438160235/4686
71:
0
72:
-5827954961669944110438277244641067365282488301844260429/140100870
73:
0
74:
34152417289221168014330073731472635186688307783087/6
75:
0
76:
-24655088825935372707687196040585199904365267828865801/30
77:
0
78:
414846365575400828295179035549542073492199375372400483487/3318
79:
0
80:
-4603784299479457646935574969019046849794257872751288919656867/230010
81:
0
82:
1677014149185145836823154509786269900207736027570253414881613/498
83:
0
84:
-2024576195935290360231131160111731009989917391198090877281083932477/3404310
85:
0
86:
660714619417678653573847847426261496277830686653388931761996983/6
87:
0
88:
-1311426488674017507995511424019311843345750275572028644296919890574047/61410
89:
0
90:
1179057279021082799884123351249215083775254949669647116231545215727922535/272118
91:
0
92:
-1295585948207537527989427828538576749659341483719435143023316326829946247/1410
93:
0
94:
1220813806579744469607301679413201203958508415202696621436215105284649447/6
95:
0
96:
-211600449597266513097597728109824233673043954389060234150638733420050668349987259/4501770
97:
0
98:
67908260672905495624051117546403605607342195728504487509073961249992947058239/6
99:
0
100:
-94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330
Definition
The Bernoulli numbers $B_n$ are the coefficients of the exponential generating functon $\frac{t}{e^t-1} = \sum_{n=0}^\infty \frac{B_nt^n}{n!}$.
Parameters
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$\sum_{n=0}^\infty \frac{B_nt^n}{n!} = \frac{t}{2}\left(\coth \frac{t}{2}-1\right)$.
(2)
$B_n = (-1)^{n+1} n \zeta(1-n)$ for $n\geq 1$, where $\zeta$ is the Riemann zeta function.
Comments
(3)
Some authors prefer to define $B^+_n = (-1)^n B_n$ as Bernoulli numbers.
Programs
(P1)
Sage
numbers = [bernoulli(n) for n in [1..10]]
Links
Data properties
Numbers are of type: rational number
both signs: True (Unknown key)