%I #35 Mar 16 2020 14:24:45

%S 1,1,1,1,1,1,0,1,1,0,-1,1,2,1,-1,0,-1,1,1,-1,0,1,-1,-1,8,-1,-1,1,0,1,

%T -1,4,4,-1,1,0,-1,1,-1,-4,8,-4,-1,1,-1,0,-1,1,-8,4,4,-8,1,-1,0,5,-5,7,

%U 4,-116,32,-116,4,7,-5,5,0,5,-5,32,-28,16,16,-28,32,-5,5,0

%N Numerators in triangle formed from Bernoulli numbers.

%C Triangle is determined by rules 0) the top number is 1; 1) each number is the sum of the two below it; 2) it is left-right symmetric; 3) the numbers in each of the border rows, after the first 3, are alternately 0.

%C Up to signs this is the difference table of the Bernoulli numbers (see A212196). The Sage script below is based on L. Seidel's algorithm and does not make use of a library function for the Bernoulli numbers; in fact it generates the Bernoulli numbers on the fly. - _Peter Luschny_, May 04 2012

%H Fabien Lange and Michel Grabisch, <a href="http://dx.doi.org/10.1016/j.disc.2008.12.007">The interaction transform for functions on lattices</a> Discrete Math. 309 (2009), no. 12, 4037-4048. [From _N. J. A. Sloane_, Nov 26 2011]

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The computation and asymptotics of the Bernoulli numbers</a>.

%H Ludwig Seidel, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1877_0157-0187.pdf">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [_Peter Luschny_, May 04 2012]

%F T(n, 0) = (-1)^n*Bernoulli(n), T(n, k) = T(n-1, k-1) - T(n, k-1) for k=1..n.

%F T(n,k) = Sum_{j=0..k} binomial(k,j)*Bernoulli(n-j). [Lange and Grabisch]

%e Triangle of fractions begins

%e 1;

%e 1/2, 1/2;

%e 1/6, 1/3, 1/6;

%e 0, 1/6, 1/6, 0;

%e -1/30, 1/30, 2/15, 1/30, -1/30;

%e 0, -1/30, 1/15, 1/15, -1/30, 0;

%e 1/42, -1/42, -1/105, 8/105, -1/105, -1/42, 1/42;

%e 0, 1/42, -1/21, 4/105, 4/105, -1/21, 1/42, 0;

%e -1/30, 1/30, -1/105, -4/105, 8/105, -4/105, -1/105, 1/30, -1/30;

%p nmax:=11; for n from 0 to nmax do T(n, 0):= (-1)^n*bernoulli(n) od: for n from 1 to nmax do for k from 1 to n do T(n, k) := T(n-1, k-1) - T(n, k-1) od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od: seq(seq(numer(T(n, k)), k=0..n), n=0..nmax); # _Johannes W. Meijer_, Jun 29 2011, revised Nov 25 2012

%t t[n_, 0] := (-1)^n*BernoulliB[n]; t[n_, k_] := t[n, k] = t[n-1, k-1] - t[n, k-1]; Table[t[n, k] // Numerator, {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 07 2014 *)

%o (Sage)

%o def BernoulliDifferenceTable(n) :

%o def T(S, a) :

%o R = [a]

%o for s in S :

%o a -= s

%o R.append(a)

%o return R

%o def M(A, p) :

%o R = T(A,0)

%o S = add(r for r in R)

%o return -S / (2*p+3)

%o R = [1/1]

%o A = [1/2,-1/2]; R.extend(A)

%o for k in (0..n-2) :

%o A = T(A,M(A,k)); R.extend(A)

%o A = T(A,0); R.extend(A)

%o return R

%o def A085737_list(n) : return [numerator(q) for q in BernoulliDifferenceTable(n)]

%o # _Peter Luschny_, May 04 2012

%Y Cf. A085738, A212196. See A051714/A051715 for another triangle that generates the Bernoulli numbers.

%K sign,frac,tabl

%O 0,13

%A _N. J. A. Sloane_, following a suggestion of _J. H. Conway_, Jul 23 2003

%E Sign flipped in formula by _Johannes W. Meijer_, Jun 29 2011