A285863 -id:A285863 - cp's OEIS frontend

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

Offset: 0

Wolfdieter Lang, Apr 28 2017

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.

			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)

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.

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

Cf. A144845, A157811.

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

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

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

# 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

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.

A285864 Triangle read by rows: a(n,m) = numerator(binomial(n,m)2^(n-m)B(n-m)) with B(k) the Bernoulli numbers A027641(k)/A027642(k).

Original entry on oeis.org

1, -1, 1, 2, -2, 1, 0, 2, -3, 1, -8, 0, 4, -4, 1, 0, -8, 0, 20, -5, 1, 32, 0, -8, 0, 10, -6, 1, 0, 32, 0, -56, 0, 14, -7, 1, -128, 0, 128, 0, -112, 0, 56, -8, 1, 0, -384, 0, 128, 0, -336, 0, 24, -9, 1, 2560, 0, -384, 0, 320, 0, -112, 0, 30, -10, 1

Offset: 0

Wolfdieter Lang, May 03 2017

The denominator triangle b(n,m) is given in A285865.
a(n,m)/b(n,m) = B(2;n,m) is the d = 2 instance of the fractional d-family of triangles B(d;n,m) = binomial(n,m)*d^(n-m)*B(n-m), for d >= 1. They are the coefficient triangles of generalized Bernoulli polynomials PB(d;n,x) = Sum_{m=0..n} B(d;n,m)*x^m for n >= 0.
{PB(d;n,x)}{n>=0} has e.g.f. EB(d;x,z) := Sum{n>=0} PB(d;n,x)*z^n = d*z*exp(x*z)/(exp(d*z)-1). B(d;n,m) is a Sheffer triangle of the Appell type for each d, denoted by (d*z/(exp(d*z - 1)), z).
PB(d;n,x) gives a (trivial) generalization of the Bernoulli polynomials with coefficients given in A196838/A196839 (rising powers of x), and this is PB(1;n,x).
The polynomials PB(d;n,x) appear in the generalized Faulhaber formula for sums of powers of arithmetic progressions SP(n,m) := Sum_{j=0..m} (a + d*j)^n, n >= 0, m >= 0, d >= 1, a = 0 for d = 1 and a from the smallest positive restricted residue system modulo d >= 2. For this Faulhaber formula see a comment in A285863, where they are named B(d;n,x).
The row sums of the rational triangle B(2;n,m) give A157779(n)/A141459(n). The alternating row sums are given in A285866/A141459(n).

			The triangle a(n,m) begins:
n\m    0    1    2   3    4    5    6  7  8   9 10 ...
0:     1
1:    -1    1
2:     2   -2    1
3:     0    2   -3   1
4:    -8    0    4  -4    1
5:     0   -8    0  20   -5    1
6:    32    0   -8   0   10   -6    1
7:     0   32    0 -56    0   14   -7  1
8:  -128    0  128   0 -112    0   56 -8  1
9:     0 -384    0 128    0 -336    0 24 -9   1
10: 2560    0 -384   0  320    0 -112  0 30 -10  1
...
The rational triangle B(2;n,m) = a(n,m)/A285865(n,m) begins:
n\m     0       1        2     3     4      5     6    7    8   9  10 ...
0:      1
1:     -1       1
2:     2/3     -2        1
3:      0       2       -3     1
4:    -8/15     0        4    -4     1
5:      0     -8/3       0   20/3   -5      1
6:    32/21     0       -8     0    10     -6     1
7:      0     32/3       0  -56/3    0     14    -7    1
8:  -128/15     0      128/3   0  -112/3    0   56/3  -8    1
9:      0    -384/5      0    128    0   -336/5   0   24   -9   1
10:  2560/33    0      -384    0    320     0   -112   0   30 -10   1
...

Cf. A027641/A027642, A157779/A141459, A196838/A196839, A285863, A285865.

T := d -> (n,m) -> numer(binomial(n, m)*d^(n-m)*bernoulli(n-m)):
for n from 0 to 10 do seq(T(2)(n,k),k=0..n) od; # Peter Luschny, May 04 2017

T[n_, m_]:=Numerator[Binomial[n, m]*2^(n - m)*BernoulliB[n - m]]; Table[T[n, m], {n, 0, 20}, {m, 0, n}] // Flatten (* Indranil Ghosh, May 06 2017 *)

T(n, m) = numerator(binomial(n, m)*2^(n - m)*bernfrac(n - m));
for(n=0, 20, for(m=0, n, print1(T(n, m),", ");); print();) \\ Indranil Ghosh, May 06 2017

from sympy import binomial, bernoulli
def T(n, m): return (binomial(n, m) * (-2)**(n - m) * bernoulli(n - m)).numerator
for n in range(21): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, May 06 2017

a(n,m) = numerator(binomial(n, m)*2^(n-m)*B(n-m)), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).

E.g.f.s of the rational column sequences {B(2;n, m)}_{n>=0} are Ecol(m, x) = (2*x/(exp(2*x) - 1))*x^m/m! (Sheffer property). Here the numerators of column m are numerator([x^m/m!] Ecol(m, x)), m >= 0.

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

A285068 Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).

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A285864 Triangle read by rows: a(n,m) = numerator(binomial(n,m)*2^(n-m)*B(n-m)) with B(k) the Bernoulli numbers A027641(k)/A027642(k).

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A285864 Triangle read by rows: a(n,m) = numerator(binomial(n,m)2^(n-m)B(n-m)) with B(k) the Bernoulli numbers A027641(k)/A027642(k).