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

%I A240581 #22 Jun 23 2014 16:31:14
%S A240581 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,
%T A240581 -116,-4,7,5,32,28,16,-16,-28,-32,-5,-2663,-388,2524,5072,6112,5072,
%U A240581 2524,-388,-2663,-691,-10264,-10652,-8128,-3056,3056,8128,10652,10264,691
%N A240581 Array read by antidiagonals: numerators of the core of the classical Bernoulli numbers.
%C A240581 Sum of antidiagonals: 2/15, 0, -2/21, 0, 2/15, 0, -10/33, 0, 1382/1365,... =-4*A164555(n+4)/A027642(n+4).
%D A240581 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.
%H A240581 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The computation and asymptotics of the Bernoulli numbers</a>.
%e A240581 As a triangle:
%e A240581 2,
%e A240581 -1,  1,
%e A240581 -1, -8, -1,
%e A240581 1,   4, -4, -1,
%e A240581 -1,  4,  8,  4, -1,
%e A240581 etc.
%p A240581 DifferenceTableBernoulli := proc(n) local A,m,k; A := array(0..n,0..n);
%p A240581 # pritty print
%p A240581 for m from 0 to n do for k from 0 to n do A[m, k] := '~' od od;
%p A240581 # compute elements
%p A240581 for m from 0 to n do A[m,0] := bernoulli(m, 1);
%p A240581    for k from m-1 by -1 to 0 do
%p A240581       A[k, m-k] := A[k+1, m-k-1] - A[k, m-k-1] od od;
%p A240581 convert(A, matrix) end:
%p A240581 A := DifferenceTableBernoulli(13); L := NULL;
%p A240581 for n from 0 to 9 do for k from 0 to n do
%p A240581    L := L, numer(A[3+k, 3+n-k]) od od;
%p A240581 L; # _Peter Luschny_, Apr 12 2014
%t A240581 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 *)
%Y A240581 Cf. A239315 (denominators).
%K A240581 sign,tabl,frac
%O A240581 0,1
%A A240581 _Paul Curtz_, Apr 08 2014
%E A240581 a(9) corrected by _Wesley Ivan Hurt_, Apr 08 2014
%E A240581 a(4) corrected by _Jean-François Alcover_, Apr 11 2014