A285068 -id:A285068 - cp's OEIS frontend

A285863 Numerators of Bernoulli numbers 3^n*B(n), with B(n) = A027641(n)/A027642(n).

1, -3, 3, 0, -27, 0, 243, 0, -2187, 0, 98415, 0, -122408577, 0, 11160261, 0, -51899996619, 0, 5664991530321, 0, -202943637014337, 0, 8938507796555139, 0, -22252066887294301257, 0, 7246946747292751629, 0, -181103830292539169071623

Offset: 0

Wolfdieter Lang, Apr 29 2017

The denominators are given in A285068.
In general the numbers B(d;n) = d^n*B(n), for n >= 0, have e.g.f. d*x/(exp(d*x) - 1). They are also the exponential convolution of the generalized Bernoulli numbers B[d,a](n), obtained from the generalized Stirling2 numbers S2[d,a], with the sequence {(-a)^n}_{n>=0}. See a comment in A157817 for the  B[4,1] and B[4,3] examples.
These numbers B(d;n) and their polynomials B(d;n,x) = Sum_{m=0..n} binomial(n, m)*B(d;n-m)*x^m are used in the generalized so-called Faulhaber formula for the sums of powers of arithmetic progressions defined by SP(d,a;n,m) := Sum_{j=0..m} (a + d*j)^n = Sum_{k=0..n} binomial(n, k)*a^(n-k)*d^k*SP(k,m) with SP(k,m) = SP(1,0;k,m), n >= 0, m >= 0, and 0^0 := 1.
The Faulhaber formula is: SP(d,a;n,m) = (1/(d*(n+1)))*[B(d;n+1,x = a+d*(m+1)) - B(d;n+1,x = d) - B(d;n+1,x = a)  + B(d;n+1,x=0) + d^(n+1)*[n=0]]. Here [n=0] is the Kronecker delta_{n,0} symbol: 1 if n=0 and 0 otherwise.
A simpler version of the Faulhaber formula is for a=0: SP(d,0;0,m) = m+1 and  SP(d,0;n,m) = d^n*(1/(n+1))*(B(n+1, x = m+1) - B(n+1, x=1)) for n >= 1, and for a an integer >= 1: Sum_{k=0..n} binomial(n, k)*a^(n-k) * d^k * (1/(k+1)) * (B(k+1, x=m+1) - B(k+1, x=1)). Here B(n, x) =  B(1;n,x) are the usual Bernoulli polynomials from A196838/A196839 or A053382/A053383.

Indranil Ghosh, Table of n, a(n) for n = 0..300
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Cf. A027641/A027642, A144845, A285068.

seq(numer(3^n*bernoulli(n)), n=0..28); # Peter Luschny, Jul 17 2017

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

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

from sympy import bernoulli
def a(n): return -3 if n == 1 else (3**n * bernoulli(n)).numerator
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 18 2017

a(n) = numerator(r(n)) with r(n) = 3^n*A027641(n)/A027642(n), n >= 0.

E.g.f. {r(n)}_{n>=0}: 3*x/(exp(3*x) - 1).

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

N. J. A. Sloane, Nov 08 2009

This sequence gives also the numerators of the generalized Bernoulli numbers B[3,1](n) = 3^n*Bernoulli(n, 1/3) with denominators given by A285068. See the formula and example section there for the rationals. The numbers B[3,2](n) = 3^n*Bernoulli(n, 2/3) = (-1)^n*B[3,1](n) have numerators (-1)^n*a(n) and denominators A285068 (proof from the e.g.f.s). - Wolfdieter Lang, Apr 28 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..250

For denominators see A157800, A285068.

Table[Numerator[BernoulliB[n, 1/3]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)

a(n) = my(x=1/3); numerator(eval(bernpol(n))); \\ Ruud H.G. van Tol, May 10 2024

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

N. J. A. Sloane, Nov 08 2009

			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)

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Peter Luschny, Generalized Bernoulli numbers.

For denominators see A157800.

The denominators of the generalized Bernoulli numbers are A285068.

Table[Numerator[BernoulliB[n, -2/3]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)

# 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

A288872 Denominators for generalized Bernoulli numbers B[5,j](n), for j=1..4, n >= 0.

Original entry on oeis.org

1, 2, 6, 1, 6, 1, 42, 1, 6, 1, 66, 1, 546, 1, 6, 1, 102, 1, 798, 1, 66, 1, 138, 1, 546, 1, 6, 1, 174, 1, 14322, 1, 102, 1, 6, 1, 383838, 1, 6, 1, 2706, 1, 1806, 1, 138, 1, 282, 1, 9282, 1, 66, 1, 318, 1, 798, 1, 174, 1, 354, 1, 11357346, 1, 6, 1, 102, 1, 64722, 1, 6, 1, 4686

Offset: 0

Wolfdieter Lang, Jul 05 2017

See, e.g., A157871 for details on B[d,a](n) with gcd(d,a) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..10101
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Cf. A027642 (denominators B[1,0]), A141459 (denominators B[2,1]), A285068 (denominators B[3,1] and B[3,2]), A141459 (denominators B[4,1] and B[4,3]).
For the numerators of B[5,j](n), for j=1..4, see A157866(n), A157883(n), (-1)^n*A157883(n), (-1)^n*A157866(n), respectively.
Cf. A157871.

Table[Denominator[BernoulliB[n, 1/5]]/5^n, {n, 0, 70}] (* Jean-François Alcover, Sep 24 2018, from PARI *)

a(n)=denominator(subst(bernpol(n, x), x, 1/5))/5^n; \\ Michel Marcus, Jul 06 2017

from sympy import bernoulli
def a(n): return bernoulli(n, 1/Integer(5)).denominator//(5**n)
print([a(n) for n in range(41)]) # Indranil Ghosh, Jul 06 2017

A285866 a(n) = numerator((-2)^nSum_{k=0..n} binomial(n,k) Bernoulli(k, 1/2)).

Original entry on oeis.org

1, -2, 11, -6, 127, -10, 221, -14, 367, -18, -1895, -22, 1447237, -26, -57253, -30, 118526399, -34, -5749677193, -38, 91546283957, -42, -1792042789427, -46, 1982765468376757, -50, -286994504449237, -54, 3187598676787485443, -58, -4625594554880206360895, -62

Offset: 0

Wolfdieter Lang, May 03 2017

Previous name: Numerators of alternating row sums of the rational triangle B2 = A285864/A285865.

The denominators are given in A141459.

Cf. A027641/A027642, A141459, A285864/A285865.

See also A157781/A157782 and A157811/A285068.

a := n -> numer((-2)^n*add(binomial(n,k)*bernoulli(k,1/2), k=0..n)):
seq(a(n), n=0..31); # Peter Luschny, Jul 24 2020

a[n_] := (-2)^n Sum[Binomial[n, k] BernoulliB[k, 1/2], {k, 0, n}] // Numerator;
Table[a[n], {n, 0, 31}] (* Peter Luschny, Jul 24 2020 *)

# uses [gen_bernoulli_number from A157811]
print([numerator((-1)^n*gen_bernoulli_number(n, 2)) for n in range(33)]) # Peter Luschny, Mar 26 2021

a(n) = numerator(Sum_{m=0..n} (-1)^m*A285864(n, m)/A285865(n, m)), n >= 0, where the rational triangle is B2(n, m) = binomial(m, m)*2^(n-m)*B(n-m), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).

More terms from Indranil Ghosh, May 06 2017

New name by Peter Luschny, Jul 24 2020

A288873 Numerators of scaled Bernoulli numbers 4^n*B(n), with B(n) = A027641(n)/A027642(n).

Original entry on oeis.org

1, -2, 8, 0, -128, 0, 2048, 0, -32768, 0, 2621440, 0, -5796528128, 0, 939524096, 0, -7767448354816, 0, 1507258642989056, 0, -95993412418797568, 0, 7516375836686024704, 0, -33265288504730187726848, 0, 19259875741830735724544, 0, -855664510723636131971203072, 0, 4966694343692730467779807805440

Offset: 0

Wolfdieter Lang, Jul 05 2017

The denominators seem to be given in A141459.

See A285863 for comments on B(d;n) = d^n*B(n), for n >= 0, with e.g.f. d*x/(exp(d*x) - 1).

			The rationals r(n) begin: 1, -2, 8/3, 0, -128/15, 0, 2048/21, 0, -32768/15, 0, 2621440/33, 0, -5796528128/1365, 0, 939524096/3, 0, -7767448354816/255, 0, 1507258642989056/399, 0, -95993412418797568/165, ...

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

Cf. A141459, A027641/A027642, (-1)^n*A239275(n)/A141459(n) (B(2;n)), A285863/A285068 (B(3;n)).

seq(numer(4^n*bernoulli(n)),n=0..28); # Peter Luschny, Jul 17 2017

Table[4^n BernoulliB[n] // Numerator, {n, 0, 30}] (* Jean-François Alcover, Jul 14 2018 *)

a(n) = numerator(4^n*bernfrac(n)); \\ Michel Marcus, Jul 06 2017

from sympy import bernoulli
def a(n): return -2 if n == 1 else (4**n * bernoulli(n)).numerator
[a(n) for n in range(31)]  # Indranil Ghosh, Jul 06 2017

a(n) = numerator(r(n)), with the rationals r(n) = 4^n*A027641(n)/A027642(n), n >= 0.

E.g.f. of {r(n)}_{n>=0}: 4*x/(exp(4*x) - 1).

cp's OEIS Frontend

A285863 Numerators of Bernoulli numbers 3^n*B(n), with B(n) = A027641(n)/A027642(n).

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A157799 Numerator of Bernoulli(n, 1/3).

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A157811 Numerator of Bernoulli(n, -2/3).

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A288872 Denominators for generalized Bernoulli numbers B[5,j](n), for j=1..4, n >= 0.

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A285866 a(n) = numerator((-2)^n*Sum_{k=0..n} binomial(n,k) * Bernoulli(k, 1/2)).

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A288873 Numerators of scaled Bernoulli numbers 4^n*B(n), with B(n) = A027641(n)/A027642(n).

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A285866 a(n) = numerator((-2)^nSum_{k=0..n} binomial(n,k) Bernoulli(k, 1/2)).