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A240581 Array read by antidiagonals: numerators of the core of the classical Bernoulli numbers. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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).

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

Table of n, a(n) for n=0..54.

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

EXAMPLE

As a triangle:

2,

-1,  1,

-1, -8, -1,

1,   4, -4, -1,

-1,  4,  8,  4, -1,

etc.

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 *)

CROSSREFS

Cf. A239315 (denominators).

Sequence in context: A172970 A172971 A306702 * A329043 A329042 A171660

Adjacent sequences:  A240578 A240579 A240580 * A240582 A240583 A240584

KEYWORD

sign,tabl,frac

AUTHOR

Paul Curtz, Apr 08 2014

EXTENSIONS

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

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

STATUS

approved

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Last modified October 29 18:53 EDT 2020. Contains 338067 sequences. (Running on oeis4.)