

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
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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*nmax1}]; diff = Table[ Differences[bb, n], {n, 1, nmax}]; A190339 = diff // Diagonal // Denominator; Table[ Table[ Take[ A190339, n], {2}], {n, 1, nmax}] // Flatten (* JeanFrançois 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



