cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Paul Curtz, Mar 15 2014

Keywords

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 *)

A235774 Let b(k) = A164555(k)/A027642(k), the sequence of "original" Bernoulli numbers with -1 instead of A164555(0)=1; then a(n) = numerator of the n-th term of the binomial transform of the b(k) sequence.

Original entry on oeis.org

-1, -1, 1, 1, 59, 3, 169, 5, 179, 7, 533, 9, 26609, 11, 79, 13, 3523, 15, 56635, 17, -168671, 19, 857273, 21, -236304031, 23, 8553247, 25, -23749438409, 27, 8615841677021, 29, -7709321025917, 31, 2577687858559, 33, -26315271552988224913
Offset: 0

Views

Author

Paul Curtz, Jan 15 2014

Keywords

Comments

(a(n)/A027642(n)) = -1, -1/2, 1/6, 1, 59/30, 3, 169/42, 5, 179/30, 7, 533/66, 9,.. .
Difference table for a(n)/A027642(n):
-1, -1/2, 1/6, 1, 59/30, 3, 169/42, ...
1/2, 2/3, 5/6, 29/30, 31/30, 43/42, 41/42, ... = A165161(n)/A051717(n+1)
1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, ... not in the OEIS
0, -1/30, -1/15, -8/105, -4/105, 4/105, 8/105, ... etc.
Compare with the array in A190339.

Crossrefs

Cf. A164558/A027642, A165161(n)/A051717(n+1).

Programs

  • Mathematica
    b[0] = -1; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}] // Numerator; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 30 2014 *)

Formula

(a(n+1) - a(n))/A027642(n) = A165161(n)/A051717(n+1).
(A164558(n) - a(n))/A027642(n) = 2's = A007395.
(a(n) - A164555(n))/A027642(n) = n - 2 = A023444(n).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.