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

Author

Paul Curtz, Apr 21 2013

Keywords

• nonn
• frac
• tabf

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).

Examples

			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;
		

Crossrefs

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

Programs

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 *)

Formula

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