Bernoulli numbers are used in some series expansions of several functions   (trigonometric, hyperbolic, gamma, etc.),   and are extremely important in number theory and analysis.

Note that there are two definitions of Bernoulli numbers;   this task will be using the modern usage   (as per   The National Institute of Standards and Technology convention).

The   nth   Bernoulli number is expressed as   Bn.



Task
   show the Bernoulli numbers   B0   through   B60.
   suppress the output of values which are equal to zero.   (Other than   B1 , all   odd   Bernoulli numbers have a value of zero.)
   express the Bernoulli numbers as fractions  (most are improper fractions).
   the fractions should be reduced.
   index each number in some way so that it can be discerned which Bernoulli number is being displayed.
   align the solidi   (/)   if used  (extra credit).


An algorithm
The Akiyama–Tanigawa algorithm for the "second Bernoulli numbers" as taken from wikipedia is as follows:

 for m from 0 by 1 to n do
    A[m] ← 1/(m+1)
    for j from m by -1 to 1 do
      A[j-1] ← j×(A[j-1] - A[j])
  return A[0] (which is Bn)

See also
 Sequence A027641 Numerator of Bernoulli number B_n on The On-Line Encyclopedia of Integer Sequences.
 Sequence A027642 Denominator of Bernoulli number B_n on The On-Line Encyclopedia of Integer Sequences.
 Entry Bernoulli number on The Eric Weisstein's World of Mathematics (TM).
 Luschny's The Bernoulli Manifesto for a discussion on    B1   =   -½   versus   +½. 


IN: scratchpad
    [
      0  1 1 "%2d : %d / %d\n" printf
      1 -1 2 "%2d : %d / %d\n" printf
      30 iota [
        1 + 2 * dup bernoulli [ numerator ] [ denominator ] bi
        "%2d : %d / %d\n" printf
      ] each
    ] time
 0 : 1 / 1
 1 : -1 / 2
 2 : 1 / 6
 4 : -1 / 30
 6 : 1 / 42
 8 : -1 / 30
10 : 5 / 66
12 : -691 / 2730
14 : 7 / 6
16 : -3617 / 510
18 : 43867 / 798
20 : -174611 / 330
22 : 854513 / 138
24 : -236364091 / 2730
26 : 8553103 / 6
28 : -23749461029 / 870
30 : 8615841276005 / 14322
32 : -7709321041217 / 510
34 : 2577687858367 / 6
36 : -26315271553053477373 / 1919190
38 : 2929993913841559 / 6
40 : -261082718496449122051 / 13530
42 : 1520097643918070802691 / 1806
44 : -27833269579301024235023 / 690
46 : 596451111593912163277961 / 282
48 : -5609403368997817686249127547 / 46410
50 : 495057205241079648212477525 / 66
52 : -801165718135489957347924991853 / 1590
54 : 29149963634884862421418123812691 / 798
56 : -2479392929313226753685415739663229 / 870
58 : 84483613348880041862046775994036021 / 354
60 : -1215233140483755572040304994079820246041491 / 56786730
Running time: 0.00489444 seconds