A285068
Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).
Original entry on oeis.org
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
Offset: 0
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)
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Table[Denominator[3^n*BernoulliB[n, 1/3]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)
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a(n) = denominator(3^n * bernfrac(n)); \\ Ruud H.G. van Tol, Jan 31 2024
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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
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# 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
A157799
Numerator of Bernoulli(n, 1/3).
Original entry on oeis.org
1, -1, -1, 1, 13, -5, -121, 49, 1093, -809, -49205, 20317, 61203943, -722813, -5580127, 34607305, 25949996501, -2145998417, -2832495743227, 167317266613, 101471818419863, -16020403322021, -4469253897850313, 1848020950359841, 11126033443528968583, -252778977216700025
Offset: 0
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Table[Numerator[BernoulliB[n, 1/3]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)
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a(n) = my(x=1/3); numerator(eval(bernpol(n))); \\ Ruud H.G. van Tol, May 10 2024
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from sympy import bernoulli, Integer
def a(n): return bernoulli(n, 1/Integer(3)).numerator # Indranil Ghosh, May 01 2017
A157811
Numerator of Bernoulli(n, -2/3).
Original entry on oeis.org
1, -7, 23, -35, 973, -245, 7943, -1295, 31813, -7721, 288715, -13475, 128296423, -882557, -4891999, 33870025, 26217383381, -2149340753, -2830613025019, 167302324405, 101475278720663, -16020469382309, -4469247530896841, 1848020660952865, 11126033993150564743
Offset: 0
From _Peter Luschny_, Mar 26 2021: (Start)
The rational numbers given in the definition start:
1, -7/6, 23/18, -35/27, 973/810, -245/243, 7943/10206, -1295/2187, 31813/65610, -7721/19683, 288715/1299078, -13475/177147, 128296423/483611310, ...
The generalized Bernoulli numbers defined 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, ... The denominators of these numbers are in A285068. (End)
The denominators of the generalized Bernoulli numbers are
A285068.
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Table[Numerator[BernoulliB[n, -2/3]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)
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# Generalized Bernoulli polynomials
def gen_bernoulli_polynomial(n, m, x):
p = sum(sum(sum(((-1)^(n-v)/(j+1))*binomial(n,k)*binomial(j,v)*(m*(v-x))^k
for v in (0..j)) for j in (0..k)) for k in (0..n))
return expand(p)
# Generalized Bernoulli numbers
def gen_bernoulli_number(n, m): return gen_bernoulli_polynomial(n, m, 1)
print([numerator((-1)^n*gen_bernoulli_number(n, 3)) for n in range(23)]) # Peter Luschny, Mar 26 2021
A157801
Numerator of Bernoulli(n, -1/3).
Original entry on oeis.org
1, -5, 11, -10, 133, -10, 131, -70, 1333, 782, -48545, -20350, 61236703, 722774, -5580043, -34607350, 25950004661, 2145998366, -2832495728863, -167317266670, 101471818426463, 16020403321958, -4469253897847277, -1848020950359910, 11126033443529034103, 252778977216699950
Offset: 0
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