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 A085738 Denominators in triangle formed from Bernoulli numbers. 14

%I

%S 1,2,2,6,3,6,1,6,6,1,30,30,15,30,30,1,30,15,15,30,1,42,42,105,105,105,

%T 42,42,1,42,21,105,105,21,42,1,30,30,105,105,105,105,105,30,30,1,30,

%U 15,105,105,105,105,15,30,1,66,66,165,165,1155,231,1155,165,165,66,66

%N Denominators 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

%D 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. [From 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>.

%F T(n, 0) = (-1)^n*Bernoulli(n); T(n, k) = T(n-1, k-1) - T(n, k-1) for k=1..n. [Corrected (sign flipped) by _R. J. Mathar_, Jun 02 2010]

%e Triangle 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

%t t[n_, 0] := (-1)^n BernoulliB[n];

%t t[n_, k_] := t[n, k] = t[n-1, k-1] - t[n, k-1];

%t Table[t[n, k] // Denominator, {n, 0, 10}, {k, 0, n}] (* _Jean-François Alcover_, Jun 04 2019 *)

%o (Sage) # BernoulliDifferenceTable is defined in A085737.

%o def A085738_list(n) : return map(denominator,BernoulliDifferenceTable(n))

%o A085738_list(6)

%o # _Peter Luschny_, May 04 2012

%Y Cf. A085737, A212196. See A051714/A051715 for another triangle that generates the Bernoulli numbers.

%K nonn,frac,tabl

%O 0,2

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

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Last modified October 20 20:04 EDT 2019. Contains 328270 sequences. (Running on oeis4.)