This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A285068 #55 Mar 10 2025 11:12:13
%S A285068 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,
%T A285068 1,290,1,4774,1,170,1,2,1,639730,1,2,1,4510,1,602,1,230,1,94,1,15470,
%U A285068 1,22
%N A285068 Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).
%C A285068 The numerators are given in A157799.
%C A285068 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).
%C A285068 This sequence gives also the denominators of {3^n*B(n)}_{n>=0} with numerators given in A285863.
%H A285068 Indranil Ghosh, <a href="/A285068/b285068.txt">Table of n, a(n) for n = 0..500</a>
%H A285068 Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%H A285068 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedBernoulliNumbers.html">Generalized Bernoulli numbers</a>.
%F A285068 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.
%F A285068 a(n) = A157800(n)/3^n, n >= 0.
%e A285068 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, ...
%e A285068 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, ...
%e A285068 From _Peter Luschny_, Mar 26 2021: (Start)
%e A285068 The generalized Bernoulli numbers as given in the Luschny link are different.
%e A285068 1, -7/2, 23/2, -35, 973/10, -245, 7943/14, -1295, 31813/10, -7721, 288715/22, -13475, 128296423/910, -882557, -4891999/2, 33870025, ...
%e A285068 The numerators of these numbers are in A157811. (End)
%t A285068 Table[Denominator[3^n*BernoulliB[n, 1/3]], {n, 0, 100}] (* _Indranil Ghosh_, Jul 18 2017 *)
%o A285068 (Python)
%o A285068 from sympy import bernoulli, Rational
%o A285068 def a(n):
%o A285068 return (3**n * bernoulli(n, Rational(1,3))).as_numer_denom()[1]
%o A285068 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jul 18 2017
%o A285068 (SageMath) # uses [gen_bernoulli_number from A157811]
%o A285068 print([denominator((-1)^n*gen_bernoulli_number(n, 3)) for n in range(23)])
%o A285068 # _Peter Luschny_, Mar 26 2021
%o A285068 (PARI) a(n) = denominator(3^n * bernfrac(n)); \\ _Ruud H.G. van Tol_, Jan 31 2024
%Y A285068 Cf. A053382/A053383, A157799, A157800, A196838/A196839, A282629, A284861, A285863.
%Y A285068 Cf. A144845, A157811.
%K A285068 nonn,easy,frac
%O A285068 0,2
%A A285068 _Wolfdieter Lang_, Apr 28 2017