A196839 Triangle of denominators of the coefficient of x^m in the n-th Bernoulli polynomial, 0 <= m <= n.

1, 2, 1, 6, 1, 1, 1, 2, 2, 1, 30, 1, 1, 1, 1, 1, 6, 1, 3, 2, 1, 42, 1, 2, 1, 2, 1, 1, 1, 6, 1, 6, 1, 2, 2, 1, 30, 1, 3, 1, 3, 1, 3, 1, 1, 1, 10, 1, 1, 1, 5, 1, 1, 2, 1, 66, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 1, 1, 1, 1, 6, 2, 1, 2730, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Author

Wolfdieter Lang, Oct 23 2011

Keywords

• nonn
• easy
• tabl
• frac

Comments

The numerator triangle is found under A196838.

This is the row reversed triangle A053383.

Examples

			The triangle starts with
n\m 0  1  2  3  4  5  6  7  8 ...
0:  1
1:  2  1
2:  6  1  1
3:  1  2  2  1
4: 30  1  1  1  1
5:  1  6  1  3  2  1
6: 42  1  2  1  2  1  1
7:  1  6  1  6  1  2  2  1
8: 30  1  3  1  3  1  3  1  1
...
For the start of the rational triangle A196838(n,m)/a(n,m) see the example section in A196838.
		

Links

• D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.

Crossrefs

Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

Programs

Maple

with(ListTools):with(PolynomialTools):
CoeffList := p -> CoefficientList(p, x):
Trow := n -> denom(CoeffList(bernoulli(n, x))):
Flatten([seq(Trow(n), n = 0..12)]); # Peter Luschny, Apr 10 2021

Formula

T(n,m) = denominator([x^m]Bernoulli(n,x)), n>=0, m=0..n.

E.g.f. of Bernoulli(n,x): z*exp(x*z)/(exp(z)-1).

See the Graham et al. reference given in A196838, eq. (7.80), p. 354.

T(n,m) = denominator(binomial(n,m)*Bernoulli(n-m)). - Fabián Pereyra, Mar 04 2020

Extensions

Name edited by M. F. Hasler, Mar 09 2020