OFFSET
0,3
COMMENTS
The difference table of Bernoulli(n,2) or B(n,2) = A164558(n)/A027642(n) is defined by placing the fractions in the upper row and calculating further rows as the differences of their preceding row:
1, 3/2, 13/6, 3, 119/30, ...
1/2, 2/3, 5/6, 29/30, ...
1/6, 1/6, 2/15, ...
0, -1/30, ...
-1/30, ...
etc.
In particular, the sums of the antidiagonals
1 = 1
1/2 + 3/2 = 2
1/6 + 2/3 + 13/6 = 3
0 + 1/6 + 5/6 + 3 = 4
etc. are the positive natural numbers. (This is rewritten for Bernoulli(n,3) in A157809).
We also have for Bernoulli(.,2)
B(0,2) = 1
B(0,2) + 2*B(1,2) = 4
B(0,2) + 3*B(1,2) + 3*B(2,2) = 12
B(0,2) + 4*B(1,2) + 6*B(2,2) + 4*B(3,2) = 32
etc. with right hand sides provided by A001787.
MATHEMATICA
nmax = 11; A164558 = Table[BernoulliB[n, 2], {n, 0, nmax}]; D164558 = Table[ Differences[A164558, n], {n, 0, nmax}]; Table[ D164558[[n-k+1, k+1]] // Numerator, {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2015 *)
CROSSREFS
KEYWORD
AUTHOR
Paul Curtz, Feb 03 2015
STATUS
approved