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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 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.)