A239315 Array read by antidiagonals: denominators of the core of the classical Bernoulli numbers.

15, 15, 15, 105, 105, 105, 21, 105, 105, 21, 105, 105, 105, 105, 105, 15, 105, 105, 105, 105, 15, 165, 165, 1155, 231, 1155, 165, 165, 33, 165, 165, 231, 231, 165, 165, 33, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015

Offset: 0

Author

Paul Curtz, Mar 15 2014

Keywords

• nonn
• tabl
• frac

Comments

We consider the autosequence A164555(n)/A027642(n) (see A190339(n)) and its difference table without the first two rows and the first two columns:

2/15, 1/15, -1/105, -1/21, -1/105, 1/15, 7/165, -5/33,...

-1/15, -8/105, -4/105, 4/105, 8/105, -4/165, -32/165,...

-1/105, 4/105, 8/105, 4/105, -116/1155, -28/165,...

1/21, 4/105, -4/105, -32/231, -16/231,...

-1/105, -8/105, -116/1155, 16/231,...

-1/15, -4/165, 28/165,...

7/165, 32/165,...

5/33,... etc.

This is an autosequence of the second kind.

The antidiagonals are palindromes in absolute values.

a(n) are the denominators. Multiples of 3.

Sum of odd antidiagonals: 2/15, -2/21, 2/15, -10/33, 1382/1365,... = -2*A000367(n+2)/A001897(n+2).

The sum of the even antidiagonals is A000004.

2/15, 0, -2/21,... = -4*A027641(n+4)/A027642(n+4) = -4*A164555(n)/A027642(n+4) and others.

Examples

			As a triangle:
15,
15,   15,
105, 105, 105,
21,  105, 105, 21,
105, 105, 105, 105, 105,
etc.
		

Crossrefs

Cf. A085737/A085738, A168516/A168426 (autosequence), A027641, A176327/A176289, A235774, A165161/A051717(n+1).

Programs

Mathematica

max = 12; tb = Table[BernoulliB[n], {n, 0, max}]; td = Table[Differences[tb, n][[3 ;; -1]], {n, 2, max - 1}]; Table[td[[n - k + 1, k]] // Denominator, {n, 1, max - 3}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 11 2014 *)