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

%I A224964 #31 Jan 06 2018 22:07:17
%S A224964 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 A224964 15,105,105,2,6,15,105,105,231,2,6,15,105,105,231,2,6,15,105,105,231,
%U A224964 15015,2,6,15,105,105,231,15015
%N A224964 Irregular triangle of the denominators of the unreduced fractions that lead to the second Bernoulli numbers.
%C A224964 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 A224964 B(0) =   1   = 2/2         (1 or 2/2 chosen arbitrarily)
%C A224964 B(1)         = 1/2
%C A224964 B(2) =  1/6  = 1/2 - 2/6
%C A224964 B(3) =   0   = 1/2 - 3/6
%C A224964 B(4) = -1/30 = 1/2 - 4/6 +  2/15
%C A224964 B(5) =   0   = 1/2 - 5/6 +  5/15
%C A224964 B(6) =  1/42 = 1/2 - 6/6 +  9/15 -  8/105
%C A224964 B(7) =   0   = 1/2 - 7/6 + 14/15 - 28/105
%C A224964 B(8) = -1/30 = 1/2 - 8/6 + 20/15 - 64/105 + 8/105.
%C A224964 The constant values along the columns of denominators are A190339(n).
%C A224964 With B(0)=1, B(2) = 1/2 -1/3, (reduced case), the last fraction of the B(2*n) is
%C A224964 1, -1/3, 2/15, -8/105, 8/105, ... = A212196(n)/A181131(n).
%C A224964 We can continue this method of sum of fractions yielding Bernoulli numbers.
%C A224964 Starting from 1/6 for B(2*n+2), we have:
%C A224964 B(2) = 1/6
%C A224964 B(4) = 1/6 - 3/15
%C A224964 B(6) = 1/6 - 5/15 + 20/105
%C A224964 B(8) = 1/6 - 7/15 + 56/105 - 28/105.
%C A224964 With the odd indices from 3, all these B(n) are the Bernoulli twin numbers -A051716(n+3)/A051717(n+3).
%F A224964 T(n,k) = A190339(k).
%e A224964 Triangle begins
%e A224964   2;
%e A224964   2;
%e A224964   2, 6;
%e A224964   2, 6;
%e A224964   2, 6, 15;
%e A224964   2, 6, 15;
%e A224964   2, 6, 15, 105;
%e A224964   2, 6, 15, 105;
%e A224964   2, 6, 15, 105, 105;
%e A224964   2, 6, 15, 105, 105;
%e A224964   2, 6, 15, 105, 105, 231;
%e A224964   2, 6, 15, 105, 105, 231;
%e A224964   2, 6, 15, 105, 105, 231, 15015;
%e A224964   2, 6, 15, 105, 105, 231, 15015;
%t A224964 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 A224964 Cf. A051716, A051717, A141044, A181131, A190339, A191302, A192456, A212196.
%K A224964 nonn,frac,tabf
%O A224964 0,1
%A A224964 _Paul Curtz_, Apr 21 2013