A240581 Array read by antidiagonals: numerators of the core of the classical Bernoulli numbers.

2, -1, 1, -1, -8, -1, 1, 4, -4, -1, -1, 4, 8, 4, -1, -1, -8, -4, 4, 8, 1, 7, -4, -116, -32, -116, -4, 7, 5, 32, 28, 16, -16, -28, -32, -5, -2663, -388, 2524, 5072, 6112, 5072, 2524, -388, -2663, -691, -10264, -10652, -8128, -3056, 3056, 8128, 10652, 10264, 691

Offset: 0

Author

Paul Curtz, Apr 08 2014

Keywords

• sign
• tabl
• frac

Comments

Sum of antidiagonals: 2/15, 0, -2/21, 0, 2/15, 0, -10/33, 0, 1382/1365,... =-4*A164555(n+4)/A027642(n+4).

Examples

			As a triangle:
2,
-1,  1,
-1, -8, -1,
1,   4, -4, -1,
-1,  4,  8,  4, -1,
etc.
		

References

• Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

Links

• Peter Luschny, The computation and asymptotics of the Bernoulli numbers.

Crossrefs

Cf. A239315 (denominators).

Programs

Maple

DifferenceTableBernoulli := proc(n) local A,m,k; A := array(0..n,0..n);
# pritty print
for m from 0 to n do for k from 0 to n do A[m, k] := '~' od od;
# compute elements
for m from 0 to n do A[m,0] := bernoulli(m, 1);
   for k from m-1 by -1 to 0 do
      A[k, m-k] := A[k+1, m-k-1] - A[k, m-k-1] od od;
convert(A, matrix) end:
A := DifferenceTableBernoulli(13); L := NULL;
for n from 0 to 9 do for k from 0 to n do
   L := L, numer(A[3+k, 3+n-k]) od od;
L; # Peter Luschny, Apr 12 2014

Mathematica

max = 13; tb = Table[BernoulliB[n], {n, 0, max}]; td = Table[Differences[tb, n][[3 ;; -1]], {n, 2, max - 1}]; Table[td[[n - k + 1, k]] // Numerator, {n, 1, max - 3}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 11 2014 *)

Extensions

a(9) corrected by Wesley Ivan Hurt, Apr 08 2014

a(4) corrected by Jean-François Alcover, Apr 11 2014