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Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).
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%I #49 Feb 16 2024 09:54:55

%S 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 1,290,1,4774,1,170,1,2,1,639730,1,2,1,4510,1,602,1,230,1,94,1,15470,

%U 1,22

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

%C The numerators are given in A157799.

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

%H Indranil Ghosh, <a href="/A285068/b285068.txt">Table of n, a(n) for n = 0..500</a>

%H 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 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedBernoulliNumbers.html">Generalized Bernoulli numbers</a>.

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

%e 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 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 From _Peter Luschny_, Mar 26 2021: (Start)

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

%e 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)

%t Table[Denominator[3^n*BernoulliB[n, 1/3]], {n, 0, 100}] (* _Indranil Ghosh_, Jul 18 2017 *)

%o (Python)

%o from sympy import bernoulli, Rational

%o def a(n):

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

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jul 18 2017

%o (SageMath) # uses [gen_bernoulli_number from A157811]

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

%o # _Peter Luschny_, Mar 26 2021

%o (PARI) a(n) = denominator(3^n * bernfrac(n)); \\ _Ruud H.G. van Tol_, Jan 31 2024

%Y Cf. A053382/A053383, A157799, A157800, A196838/A196839, A282629, A284861, A285863.

%Y Cf. A144845, A157811.

%K nonn,easy,frac

%O 0,2

%A _Wolfdieter Lang_, Apr 28 2017