A224964 - OEIS

%I #31 Jan 06 2018 22:07:17

%S 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,

%T 15,105,105,2,6,15,105,105,231,2,6,15,105,105,231,2,6,15,105,105,231,

%U 15015,2,6,15,105,105,231,15015

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

%C 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

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

%C B(1)         = 1/2

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

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

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

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

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

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

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

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

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

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

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

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

%C B(2) = 1/6

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

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

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

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

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

%e Triangle begins

%e   2;

%e   2;

%e   2, 6;

%e   2, 6;

%e   2, 6, 15;

%e   2, 6, 15;

%e   2, 6, 15, 105;

%e   2, 6, 15, 105;

%e   2, 6, 15, 105, 105;

%e   2, 6, 15, 105, 105;

%e   2, 6, 15, 105, 105, 231;

%e   2, 6, 15, 105, 105, 231;

%e   2, 6, 15, 105, 105, 231, 15015;

%e   2, 6, 15, 105, 105, 231, 15015;

%t 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 *)

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

%K nonn,frac,tabf

%O 0,1

%A _Paul Curtz_, Apr 21 2013