A141459 -id:A141459 - cp's OEIS frontend

A157779 Numerator of Bernoulli(n, 1/2).

1, 0, -1, 0, 7, 0, -31, 0, 127, 0, -2555, 0, 1414477, 0, -57337, 0, 118518239, 0, -5749691557, 0, 91546277357, 0, -1792042792463, 0, 1982765468311237, 0, -286994504449393, 0, 3187598676787461083, 0, -4625594554880206790555, 0, 16555640865486520478399, 0

Offset: 0

N. J. A. Sloane, Nov 08 2009

Included for completeness, normally alternating zeros like this are omitted. A001896 is the official version of this sequence.

The sequence {a(n)/A141459(n)} gives the generalized Bernoulli numbers B[2,1] obtained from the generalized Stirling2 triangle S3[2,1] = A154537. See the formula section. - Wolfdieter Lang, Apr 27 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.

For denominators see A157780 and A141459.

Numerator[BernoulliB[Range[0,40],1/2]] (* Harvey P. Dale, May 04 2013 *)

a(n) = numerator(subst(bernpol(n, x), x, 1/2)); \\ Altug Alkan, Jul 05 2016

def A157779_list(size):
    f = x / sum(x^(n*2+1)/factorial(n*2+1) for n in (0..2*size))
    t = taylor(f, x, 0, size)
    return [(factorial(n)*s).numerator() for n,s in enumerate(t.list())]
print(A157779_list(33)) # Peter Luschny, Jul 05 2016

Let P(x) = Sum_{n>=0} x^(2*n+1)/(2*n+1)!; then a(n) = numerator( n! [x^n] x/P(x) ). - Peter Luschny, Jul 05 2016
a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A154537(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A145901(n, k). The denominators are in A141459. r(n) = B[2,1](n) = 2^n*B(n, 1/2) with the Bernoulli polynomials A196838/A196839 or A053382/A053383. - Wolfdieter Lang, Apr 27 2017
a(n) = numerator(-(1-2^(1-n))*Bernoulli(n)). - Fabián Pereyra, Dec 31 2022

A157817 Numerator of Bernoulli(n, 1/4).

Original entry on oeis.org

1, -1, -1, 3, 7, -25, -31, 427, 127, -12465, -2555, 555731, 1414477, -35135945, -57337, 2990414715, 118518239, -329655706465, -5749691557, 45692713833379, 91546277357, -7777794952988025, -1792042792463, 1595024111042171723, 1982765468311237, -387863354088927172625

Offset: 0

N. J. A. Sloane, Nov 08 2009

From Wolfdieter Lang, Apr 28 2017: (Start)
The rationals r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A285061(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) define generalized Bernoulli numbers, named B[4,1](n), in terms of the generalized Stirling2 numbers S2[4,1]. The numerators of r(n) are a(n) and the denominators A141459(n). r(n) = B[4,1](n) = 4^n*B(n, 1/4) with the Bernoulli polynomials B(n, x) = Bernoulli(n, x) from A196838/A196839 or A053382/A053383.
The generalized Bernoulli numbers B[4,3](n) = Sum_{k=0..n} ((-1)^k/(k+1))* A225467(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) satisfy
  B[4,3](n) = 4^n*B(n, 3/4) = (-1)^n*B[4,1](n). They have numerators (-1)^n*a(n) and also denominators A141459(n). (End)

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

For denominators see A157818 and A141459.

Table[Numerator[BernoulliB[n, 1/4]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)

From Wolfdieter Lang, Apr 28 2017: (Start)
a(n) = numerator(Bernoulli(n, 1/4)) with denominator A157818(n) (see the name).
a(n) = numerator(4^n*Bernoulli(n, 1/4)) with denominator A141459(n) = A157818(n)/4^n.
a(n)*(-1)^n = numerator(4^n*Bernoulli(n, 3/4)) with denominator A141459(n).
(End)

A239275 a(n) = numerator(2^n * Bernoulli(n, 1)).

Original entry on oeis.org

1, 1, 2, 0, -8, 0, 32, 0, -128, 0, 2560, 0, -1415168, 0, 57344, 0, -118521856, 0, 5749735424, 0, -91546451968, 0, 1792043646976, 0, -1982765704675328, 0, 286994513002496, 0, -3187598700536922112, 0, 4625594563496048066560, 0, -16555640873195841519616, 0, 22142170101965089931264, 0

Offset: 0

Paul Curtz, Mar 13 2014

Difference table of f(n) = 2^n *A164555(n)/A027642(n) = a(n)/A141459(n):
  1,           1,      2/3,        0,    -8/15,      0,  32/21, 0,...
  0,        -1/3,     -2/3,    -8/15,     8/15,  32/21, -32/21,...
  -1/3,     -1/3,     2/15,    16/15,  104/105, -64/21,...
  0,        7/15,    14/15,   -8/105, -424/105,...
  7/15,     7/15, -106/105, -416/105,...
  0,      -31/21,   -62/31,
  -31/21, -31/21,...
  0,... etc.
Main diagonal: A212196(n)/A181131(n). See A190339(n).
First upper diagonal: A229023(n)/A181131(n).
The inverse binomial transform of f(n) is g(n). Reciprocally, the inverse binomial transform of g(n) is f(n) with -1 instead of f(1)=1, i.e., f(n) signed.
Sum of the antidiagonals: 1,1,0,-1,0,3,0,-17,... = (-1)^n*A036968(n) = -A226158(n+1).
Following A211163(n+2), f(n) is the coefficients of a polynomial in Pi^n.
Bernoulli numbers, twice, and Genocchi numbers, twice, are linked to Pi.
f(n) - g(n) = -A226158(n).
Also the numerators of the centralized Bernoulli polynomials 2^n*Bernoulli(n, x/2+1/2) evaluated at x=1. The denominators are A141459. - Peter Luschny, Nov 22 2015
(-1)^n*a(n) = 2^n*numerator(A027641(n)/A027642(n)) (that is the present sequence with a(1) = -1 instead of +1). - Wolfdieter Lang, Jul 05 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017. See B(2;n), eq. (53).

Cf. A141459 (denominators), A001896/A001897, A027641/A027642.

seq(numer(2^n*bernoulli(n, 1)), n=0..35); # Peter Luschny, Jul 17 2017

Table[Numerator[2^n*BernoulliB[n, 1]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

from sympy import bernoulli
def a(n): return (2**n * bernoulli(n, 1)).numerator
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 18 2017

a(n) = numerators of 2^n * A164555(n)/A027642(n).

Numerators of the binomial transform of A157779(n)/(interleave A001897(n), 1)(conjectured).

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

A165636 a(n) = A091137(n)/2^n.

Original entry on oeis.org

1, 1, 3, 3, 45, 45, 945, 945, 14175, 14175, 467775, 467775, 638512875, 638512875, 1915538625, 1915538625, 488462349375, 488462349375, 194896477400625, 194896477400625, 32157918771103125, 32157918771103125, 2218896395206115625, 2218896395206115625, 3028793579456347828125, 3028793579456347828125, 9086380738369043484375

Offset: 0

Paul Curtz, Sep 23 2009

nonn

Cf. A171080, A001897, A036278.

A091137 := proc(n) local a, i, p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; end if; a := a*p^floor(n/(p-1)) ; end do: a ; end proc:
A165636 := proc(n) A091137(n)/2^n ; end proc: # R. J. Mathar, Jul 07 2011

a(n)=my(p=primes(primepi(n+1)));prod(i=1,#p,p[i]^(n\(p[i]-1)))>>n \\ Charles R Greathouse IV, Jul 07 2011

a(n+1) = a(n)* A141459(n+1).

A165886 a(n) = A165641(n+1)/A165641(n).

Original entry on oeis.org

1, 6, 1, 60, 1, 42, 1, 120, 1, 66, 1, 5460, 1, 6, 1, 4080, 1, 798, 1, 660, 1, 138, 1, 10920, 1, 6, 1, 1740, 1, 14322, 1, 8160, 1, 6, 1, 3838380, 1, 6, 1, 54120, 1, 1806, 1, 1380, 1, 282, 1, 371280, 1, 66, 1, 3180, 1, 798, 1, 3480, 1, 354, 1, 113573460, 1, 6, 1, 16320, 1, 64722, 1, 60, 1, 4686, 1, 560403480, 1, 6

Offset: 0

Paul Curtz, Sep 29 2009

nonn

Conjecture: a(n)/A141459(n+1) = A006519(n+1).

The conjecture is correct at least up to n<=2000. - R. J. Mathar, Jul 04 2011

Antti Karttunen, Table of n, a(n) for n = 0..16384

A001316 := proc(n) 2^A000120(n) ;end proc:
A165641 := proc(n) A091137(n)/A001316(n) ;end proc:
A165886 := proc(n) A165641(n+1)/A165641(n) ;end proc:
seq(A165886(n),n=0..60) ; # R. J. Mathar, Jul 04 2011

A001316(n) = 2^hammingweight(n);
A091137(n) = { my(r=1); forprime(p=2, n+1, r*=p^(n\(p-1))); (r); }; \\ From A091137
A165641(n) = (A091137(n)/A001316(n));
A165886(n) = (A165641(n+1)/A165641(n)); \\ Antti Karttunen, Dec 19 2018, after Maple-program

More terms from Antti Karttunen, Dec 19 2018

A256675 Denominators of the inverse binomial transform of Bernoulli(n+2).

Original entry on oeis.org

6, 6, 15, 15, 105, 21, 105, 15, 165, 33, 15015, 1365, 1365, 3, 255, 255, 33915, 399, 21945, 165, 3795, 69, 31395, 1365, 1365, 3, 435, 435, 1038345, 7161, 608685, 255, 255, 3, 959595, 959595, 959595, 3, 6765, 6765, 2036265, 903, 103845, 345, 16215, 141, 1090635

Offset: 0

Paul Curtz, Apr 07 2015

nonn

Difference table of B(n+2):
1/6,        0,  -1/30,     0,   1/42,     0, -1/30, ...
-1/6,   -1/30,   1/30,  1/42,  -1/42, -1/30,   ...
2/15,    1/15, -1/105, -1/21, -1/105,   ...
-1/15, -8/105, -4/105, 4/105,    ...
-1/105, 4/105,  8/105,   ...
1/21,   4/105,    ...
-1/105,   ...
...
a(n) is the denominator of the n-th term of the first column.
a(n+2) is the denominator of the n-th term of the third row.
See A239315(n), which is the table without the first two rows.
Inverse binomial transform: 1/6, -1/6, 2/15, -1/15, -1/105, 1/21, -1/105, -1/15, 7/165, 5/33, -2663/15015, ... .

Cf. A190339, A239315, A029765, A001897, A141459, A172087, A168516, A177735, A256595.

max = 42; bb = Table[BernoulliB[n+2], {n, 0, max}]; dd = Table[Differences[bb, n], {n, 0, max}]; dd[[All, 1]] // Denominator (* Jean-François Alcover, Apr 09 2015 *)

lista(nn) = {A = vector(nn, n, bernfrac(n+1)); for (i=1, #A-1, for(j=0,i-1,A[i+1]-=binomial(i,j)*A[j+1])); for (i=1, #A, print1(denominator(A[i]), ", "));} \\ Michel Marcus, Apr 08 2015

a(2n) = A029765(n).
a(2n+3) = A001897(n+2).
a(2n)/a(2n+1) = A177735(n).
a(2n+4)/a(2n+3) = A177735(n+3).

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.

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

A336517 T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^nSum_{k=0..n} binomial(n, k) Bernoulli(k, 1/2) * x^(n-k). Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, -1, 0, 4, 0, -2, 0, 8, 7, 0, -8, 0, 16, 0, 14, 0, -80, 0, 32, -31, 0, 28, 0, -80, 0, 64, 0, -62, 0, 392, 0, -224, 0, 128, 127, 0, -496, 0, 1568, 0, -1792, 0, 256, 0, 762, 0, -992, 0, 9408, 0, -1536, 0, 512, -2555, 0, 1524, 0, -4960, 0, 6272, 0, -3840, 0, 1024

Offset: 0

Peter Luschny, Jul 24 2020

Consider polynomials B_a(n, x) = a^n*Sum_{k=0..n} binomial(n, k)*Bernoulli(k, 1/a)*x^(n - k), with a != 0. They form an Appell sequence, the case a = 1 are the Bernoulli polynomials. T(n, k) are the numerators of the coefficients of the polynomials in the case a = 2.

			Rational polynomials start, coefficients of [numerators | denominators]:
                                           [ [1], [ 1]]
                                       [[0,   2], [ 1, 1]]
                                   [[-1, 0,   4], [ 3, 1, 1]]
                             [[0,    -2, 0,   8], [ 1, 1, 1, 1]]
                          [[7, 0,    -8, 0,  16], [15, 1, 1, 1, 1]]
                    [[0,   14, 0,   -80, 0,  32], [ 1, 3, 1, 3, 1, 1]]
               [[-31, 0,   28, 0,   -80, 0,  64], [21, 1, 1, 1, 1, 1, 1]]
           [[0,  -62, 0,  392, 0,  -224, 0, 128], [ 1, 3, 1, 3, 1, 1, 1, 1]]
      [[127, 0, -496, 0, 1568, 0, -1792, 0, 256], [15, 1, 3, 1, 3, 1, 3, 1, 1]]
   [[0, 762, 0, -992, 0, 9408, 0, -1536, 0, 512], [ 1, 5, 1, 1, 1, 5, 1, 1, 1, 1]]

Cf. A285865 (denominators), A336454 (polynomial denominator), A141459, A157779, A285866.

Bcp := n -> 2^n*add(binomial(n,k)*bernoulli(k,1/2)*x^(n-k), k=0..n):
polycoeff := p -> seq(numer(coeff(p, x, k)), k = 0..degree(p, x)):
Trow := n -> polycoeff(Bcp(n)): seq(Trow(n), n=0..10);

Denominator(b(n, 1)) = A141459(n).
Numerator(b(n, -1)) = A285866(n).
Numerator(b(n, 0)) = A157779(n).

cp's OEIS Frontend

A157779 Numerator of Bernoulli(n, 1/2).

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A157817 Numerator of Bernoulli(n, 1/4).

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Mathematica

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A239275 a(n) = numerator(2^n * Bernoulli(n, 1)).

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Maple

Mathematica

Python

Formula

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

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Mathematica

PARI

Python

A165636 a(n) = A091137(n)/2^n.

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Maple

PARI

Formula

A165886 a(n) = A165641(n+1)/A165641(n).

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Maple

PARI

Extensions

A256675 Denominators of the inverse binomial transform of Bernoulli(n+2).

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PARI

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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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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).

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

A336517 T(n, k) = numerator([x^k] b(n, x)), where b(n, x) = 2^nSum_{k=0..n} binomial(n, k) Bernoulli(k, 1/2) * x^(n-k). Triangle read by rows, for 0 <= k <= n.