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A224964 Irregular triangle of the denominators of the unreduced fractions that lead to the second Bernoulli numbers. 0
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified April 25 13:31 EDT 2019. Contains 322461 sequences. (Running on oeis4.)