login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A085738 Denominators in triangle formed from Bernoulli numbers. 14
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

REFERENCES

Lange, Fabien; and Grabisch, Michel; The interaction transform for functions on lattices. Discrete Math. 309 (2009), no. 12, 4037-4048. [From N. J. A. Sloane, Nov 26 2011]

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]

LINKS

Table of n, a(n) for n=0..62.

Peter Luschny, The computation and asymptotics of the Bernoulli numbers.

FORMULA

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

EXAMPLE

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

PROG

(Sage) # BernoulliDifferenceTable is defined in A085737.

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

A085738_list(6)

# Peter Luschny, May 04 2012

CROSSREFS

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

Sequence in context: A277021 A275037 A174833 * A227608 A276484 A100641

Adjacent sequences:  A085735 A085736 A085737 * A085739 A085740 A085741

KEYWORD

nonn,frac,tabl

AUTHOR

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

EXTENSIONS

Flipped a sign in the formula - R. J. Mathar, Jun 02 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 18 20:32 EST 2018. Contains 299330 sequences. (Running on oeis4.)