A085738 Denominators in triangle formed from Bernoulli numbers.

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, 42, 42, 1, 42, 21, 105, 105, 21, 42, 1, 30, 30, 105, 105, 105, 105, 105, 30, 30, 1, 30, 15, 105, 105, 105, 105, 15, 30, 1, 66, 66, 165, 165, 1155, 231, 1155, 165, 165, 66, 66

Offset: 0

Author

N. J. A. Sloane following a suggestion of J. H. Conway, Jul 23 2003

Keywords

• nonn
• frac
• tabl

Comments

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.

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

Examples

			Triangle begins
    1
   1/2,   1/2
   1/6,   1/3,   1/6
    0,    1/6,   1/6,     0
  -1/30,  1/30,  2/15,   1/30,  -1/30
    0,   -1/30,  1/15,   1/15,  -1/30,     0
   1/42, -1/42, -1/105,  8/105, -1/105,  -1/42,   1/42
    0,    1/42, -1/21,   4/105,  4/105,  -1/21,   1/42,   0
  -1/30,  1/30, -1/105, -4/105,  8/105,  -4/105, -1/105, 1/30, -1/30
		

Links

• Fabien Lange and Michel Grabisch, The interaction transform for functions on lattices Discrete Math. 309 (2009), no. 12, 4037-4048. [From _N. J. A. Sloane_, Nov 26 2011]
• Peter Luschny, The computation and asymptotics of the Bernoulli numbers.
• 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. [_Peter Luschny_, May 04 2012]

Crossrefs

See A051714/A051715 for another triangle that generates the Bernoulli numbers.

Cf. A085737, A212196.

Programs

Mathematica

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] // Denominator, {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 04 2019 *)

Sage

# uses[BernoulliDifferenceTable from A085737]
def A085738_list(n): return [q.denominator() for q in BernoulliDifferenceTable(n)]
A085738_list(6)
# Peter Luschny, May 04 2012

Formula

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]

Let U(m, n) = (-1)^(m + n)*T(m+n, n). Then the e.g.f. for U(m, n) is (x - y)/(e^x - e^y). - Ira M. Gessel, Jun 12 2021