A285068 - OEIS

A285068

Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).

1, 2, 2, 1, 10, 1, 14, 1, 10, 1, 22, 1, 910, 1, 2, 1, 170, 1, 266, 1, 110, 1, 46, 1, 910, 1, 2, 1, 290, 1, 4774, 1, 170, 1, 2, 1, 639730, 1, 2, 1, 4510, 1, 602, 1, 230, 1, 94, 1, 15470, 1, 22

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OFFSET

0,2

COMMENTS

The numerators are given in A157799.

Because B(n, 2/3) = (-1)^n*B(n, 1/3) (from the e.g.f. z*exp(x*z)/(exp(z)-1) of Bernoulli polynomials {B(n, x)}_{n>=0}) one has for the numbers B[3,2](n) = 3^n*B(n, 2/3) the numerators (-1)^n*A157799(n) and the denominators a(n).

This sequence gives also the denominators of {3^n*B(n)}_{n>=0} with numerators given in A285863.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..500

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Peter Luschny, Generalized Bernoulli numbers.

FORMULA

a(n) = denominator(r(n)) with the rationals (in lowest terms) r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A282629(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A284861(n, k). r(n) = B[3,1](n) = 3^n*B(n, 1/3) with the Bernoulli polynomials A196838/A196839 or A053382/A053383.

a(n) = A157800(n)/3^n, n >= 0.

EXAMPLE

The Bernoulli numbers r(n) = B[3,1](n) begin: 1, -1/2, -1/2, 1, 13/10, -5, -121/14, 49, 1093/10, -809, -49205/22, 20317, 61203943/910, -722813, -5580127/2, 34607305, ...

The Bernoulli numbers B(3,2](n) begin: 1, 1/2, -1/2, -1, 13/10, 5, -121/14, -49, 1093/10, 809, -49205/22, -20317, 61203943/910, 722813, -5580127/2, -34607305, ...

From Peter Luschny, Mar 26 2021: (Start)

The generalized Bernoulli numbers as given in the Luschny link are different.

1, -7/2, 23/2, -35, 973/10, -245, 7943/14, -1295, 31813/10, -7721, 288715/22, -13475, 128296423/910, -882557, -4891999/2, 33870025, ...The numerators of these numbers are in A157811. (End)

MATHEMATICA

Table[Denominator[3^n*BernoulliB[n, 1/3]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

PROG

(Python)

from sympy import bernoulli, Rational

def a(n):

return (3**n * bernoulli(n, Rational(1, 3))).as_numer_denom()[1]

print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 18 2017

(SageMath) # uses [gen_bernoulli_number from A157811]

print([denominator((-1)^n*gen_bernoulli_number(n, 3)) for n in range(23)])

# Peter Luschny, Mar 26 2021

(PARI) a(n) = denominator(3^n * bernfrac(n)); \\ Ruud H.G. van Tol, Jan 31 2024