A224964 Irregular triangle of the denominators of the unreduced fractions that lead to the second Bernoulli numbers.

2, 2, 2, 6, 2, 6, 2, 6, 15, 2, 6, 15, 2, 6, 15, 105, 2, 6, 15, 105, 2, 6, 15, 105, 105, 2, 6, 15, 105, 105, 2, 6, 15, 105, 105, 231, 2, 6, 15, 105, 105, 231, 2, 6, 15, 105, 105, 231, 15015, 2, 6, 15, 105, 105, 231, 15015

Offset

0,1

Comments

The triangle of fractions A192456(n)/A191302(n) leading to the second Bernoulli numbers written in A191302(n) is the reduced case. The unreduced case is

B(0) = 1 = 2/2 (1 or 2/2 chosen arbitrarily)

B(1) = 1/2

B(2) = 1/6 = 1/2 - 2/6

B(3) = 0 = 1/2 - 3/6

B(4) = -1/30 = 1/2 - 4/6 + 2/15

B(5) = 0 = 1/2 - 5/6 + 5/15

B(6) = 1/42 = 1/2 - 6/6 + 9/15 - 8/105

B(7) = 0 = 1/2 - 7/6 + 14/15 - 28/105

B(8) = -1/30 = 1/2 - 8/6 + 20/15 - 64/105 + 8/105.

The constant values along the columns of denominators are A190339(n).

With B(0)=1, B(2) = 1/2 -1/3, (reduced case), the last fraction of the B(2*n) is

1, -1/3, 2/15, -8/105, 8/105, ... = A212196(n)/A181131(n).

We can continue this method of sum of fractions yielding Bernoulli numbers.

Starting from 1/6 for B(2*n+2), we have:

B(2) = 1/6

B(4) = 1/6 - 3/15

B(6) = 1/6 - 5/15 + 20/105

B(8) = 1/6 - 7/15 + 56/105 - 28/105.

With the odd indices from 3, all these B(n) are the Bernoulli twin numbers -A051716(n+3)/A051717(n+3).

Links

Table of n, a(n) for n=0..55.

Formula

T(n,k) = A190339(k).

Example

Triangle begins
  2;
  2;
  2, 6;
  2, 6;
  2, 6, 15;
  2, 6, 15;
  2, 6, 15, 105;
  2, 6, 15, 105;
  2, 6, 15, 105, 105;
  2, 6, 15, 105, 105;
  2, 6, 15, 105, 105, 231;
  2, 6, 15, 105, 105, 231;
  2, 6, 15, 105, 105, 231, 15015;
  2, 6, 15, 105, 105, 231, 15015;

Mathematica

nmax = 7; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[ Differences[bb, n], {n, 1, nmax}]; A190339 = diff // Diagonal // Denominator; Table[ Table[ Take[ A190339, n], {2}], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Apr 25 2013 *)

Crossrefs

Cf. A051716, A051717, A141044, A181131, A190339, A191302, A192456, A212196.

Sequence in context: A286847 A291439 A023957 * A278165 A074928 A285713

Adjacent sequences: A224961 A224962 A224963 * A224965 A224966 A224967

Keyword

nonn,frac,tabf

Author

Paul Curtz, Apr 21 2013

Status

approved