A157779 -id:A157779 - cp's OEIS frontend

A141459 a(n) = Product_{p-1 divides n} p, where p is an odd prime.

1, 1, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, 1365, 1, 3, 1, 255, 1, 399, 1, 165, 1, 69, 1, 1365, 1, 3, 1, 435, 1, 7161, 1, 255, 1, 3, 1, 959595, 1, 3, 1, 6765, 1, 903, 1, 345, 1, 141, 1, 23205, 1, 33, 1, 795, 1, 399, 1, 435, 1, 177, 1, 28393365, 1, 3, 1, 255, 1, 32361, 1, 15, 1, 2343, 1, 70050435

Offset: 0

Paul Curtz, Aug 08 2008

nonn

Previous name was: A027760(n)/2 for n>=1, a(0) = 1.
Conjecture: a(n) = denominator of integral_{0..1}(log(1-1/x)^n) dx. - Jean-François Alcover, Feb 01 2013
Define the generalized Bernoulli function as B(s,z) = -s*z^s*HurwitzZeta(1-s,1/z) for Re(1/z) > 0 and B(0,z) = 1 for all z; further the generalized Bernoulli polynomials as Bp(m,n,z) = Sum_{j=0..n} B(j,m)*C(n,j)*(z-1)^(n-j) then the a(n) are denominators of Bp(2,n,1), i. e. of the generalized Bernoulli numbers in the case m=2. The numerators of these numbers are A157779(n). - Peter Luschny, May 17 2015
From Peter Luschny, Nov 22 2015: (Start)
a(n) are the denominators of the centralized Bernoulli polynomials 2^n*Bernoulli(n, x/2+1/2) evaluated at x=1. The numerators are A239275(n).
a(n) is the odd part of A141056(n).
a(n) is squarefree, by the von Staudt-Clausen theorem. (End)
Apparently a(n) = denominator(Sum_{k=0..n-1}(-1)^k*E2(n-1, k+1)/binomial(2*n-1, k+1)) where E2(n, k) denotes the second-order Eulerian numbers A340556. - Peter Luschny, Feb 17 2021

			The denominators of 1, 0, -1/3, 0, 7/15, 0, -31/21, 0, 127/15, 0, -2555/33, 0, 1414477/1365, ...

Robert Israel, Table of n, a(n) for n = 0..10000
Peter Luschny, Generalized Bernoulli Numbers and Polynomials

Cf. A027760, A141056, A141459, A157779, A157818, A160014, A226157, A239275, A340556.

Bfun := (s,z) -> `if`(s=0,1,-s*z^s*Zeta(0,1-s,1/z): # generalized Bernoulli function
Bpoly := (m,n,z) -> add(Bfun(j,m)*binomial(n,j)*(z-1)^(n-j),j=0..n): # generalized Bernoulli polynomials
seq(Bpoly(2,n,1),n=0..50): denom([%]);
# which simplifies to:
a := n -> 0^n+add(Zeta(1-j)*(2^j-2)*j*binomial(n,j), j=1..n):
seq(denom(a(n)), n=0..50); # Peter Luschny, May 17 2015
# Alternatively:
with(numtheory):
ClausenOdd := proc(n) local S, m;
S := map(i -> i + 1, divisors(n));
S := select(isprime, S) minus {2};
mul(m, m = S) end: seq(ClausenOdd(n), n=0..72); # Peter Luschny, Nov 22 2015
# Alternatively:
N:= 1000: # to get a(0) to a(N)
V:= Array(0..N, 1):
for p in select(isprime, [seq(i,i=3..N+1,2)]) do
  R:=[seq(j,j=p-1..N, p-1)]:
  V[R]:= V[R] * p;
od:
convert(V,list); # Robert Israel, Nov 22 2015

a[n_] := If[OddQ[n], 1, Denominator[-2*(2^(n - 1) - 1)*BernoulliB[n]]]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Jan 30 2013 *)
Table[Times @@ Select[Divisors@ n + 1, PrimeQ@ # && OddQ@ # &] + Boole[n == 0], {n, 0, 72}] (* Michael De Vlieger, Apr 30 2017 *)

A141056(n) =
{
    p = 1;
    if (n > 0,
        fordiv(n, d,
            r = d + 1;
            if (isprime(r) & r>2, p = p*r)
        )
    );
    return(p)
}
for(n=0, 72, print1(A141056(n), ", ")); \\ Peter Luschny, Nov 22 2015

def A141459_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).denominator() for n,s in enumerate (t.list())]
print(A141459_list(72)) # Peter Luschny, Jul 05 2016

a(2*n+1) = 1. a(2*n)= A001897(n).
a(n) = denominator(0^n + Sum_{j=1..n} zeta(1-j)*(2^j-2)*j*C(n,j)). - Peter Luschny, May 17 2015
Let P(x)= Sum_{n>=0} x^(2*n+1)/(2*n+1)! then a(n) = denominator( n! [x^n] x/P(x) ). - Peter Luschny, Jul 05 2016
a(n) = A157818(n)/4^n. See a comment under A157817, also for other Bernoulli numbers B[4,1] and B[4,3] with this denominator. - Wolfdieter Lang, Apr 28 2017

1 prepended and offset set to 0 by Peter Luschny, May 17 2015

New name from Peter Luschny, Nov 22 2015

A157780 Denominator of Bernoulli(n, 1/2).

Original entry on oeis.org

1, 1, 12, 1, 240, 1, 1344, 1, 3840, 1, 33792, 1, 5591040, 1, 49152, 1, 16711680, 1, 104595456, 1, 173015040, 1, 289406976, 1, 22900899840, 1, 201326592, 1, 116769423360, 1, 7689065201664, 1, 1095216660480, 1, 51539607552, 1

Offset: 0

N. J. A. Sloane, Nov 08 2009

Included for completeness, A033469 is the official version of this sequence.

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

For numerators see A157779.

Table[Denominator[BernoulliB[n, 1/2]], {n, 0, 50}] (* Vincenzo Librandi, Mar 19 2014 *)

a(n) = denominator(-(1-2^(1-n))*Bernoulli(n)). - Fabián Pereyra, Dec 31 2022

A224783 Denominator of Bernoulli(n,1/2) - Bernoulli(n,0).

Original entry on oeis.org

1, 2, 4, 1, 16, 1, 64, 1, 256, 1, 1024, 1, 4096, 1, 16384, 1, 65536, 1, 262144, 1, 1048576, 1, 4194304, 1, 16777216, 1, 67108864, 1, 268435456, 1, 1073741824, 1, 4294967296, 1, 17179869184, 1, 68719476736, 1, 274877906944, 1, 1099511627776

Offset: 0

Paul Curtz, Apr 17 2013

See A157779 and A157780 for values of Bernoulli(n,1/2), and A027641 and A027642 for values of Bernoulli(n,0).
B(n,1/2) - B(n,0) = 0, 1/2, -1/4, 0, 1/16, 0, -3/64, 0, 17/256, 0, -155/1024, 0, 2073/4096, 0, -38227/16384,... for n>=0.
The sequence of numerators is 0, 1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227, 0, 929569, 0, -28820619, 0, 1109652905,...and appears to contain a mix of A001469 and A036968.

			a(0) = 1-1, a(1) = 0+1/2, a(2) = -1/12-1/6=-1/4.

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Cf. A001469, A027641, A027642, A036968, A059222, A157779, A157780.

A224783 := proc(n)
    bernoulli(n,1/2)-bernoulli(n) ;
    denom(%) ;
end proc: # R. J. Mathar, Apr 25 2013

Table[Denominator[BernoulliB[n, 1/2] - BernoulliB[n, 0]], {n, 0, 50}] (* Vincenzo Librandi, Mar 19 2014 *)

Vec((4*x^5-9*x^3-x^2+2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)) + O(x^100)) \\ Colin Barker, Mar 20 2014

a(n) = A059222(n+1) if n <> 1.
From Colin Barker, Mar 19 2014: (Start)
G.f.: (4*x^5-9*x^3-x^2+2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)).
a(n) = 5*a(n-2)-4*a(n-4) for n>5.
a(n) = (1+(-2)^n-(-1)^n+2^n)/2 for n>1. (End).

A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

Original entry on oeis.org

1, 1, -1, -1, 7, 3, -31, -17, 127, 155, -2555, -2073, 1414477, 38227, -57337, -929569, 118518239, 28820619, -5749691557, -1109652905, 91546277357, 51943281731, -1792042792463, -2905151042481, 1982765468311237, 191329672483963, -286994504449393, -14655626154768697, 3187598676787461083, 1291885088448017715, -4625594554880206790555

Offset: 0

Paul Curtz, Jul 30 2013

sign

a(n) is the numerators of numbers derived from Bernoulli and Genocchi numbers. The denominators b(n) are the Clausen numbers A141056.
The numbers are
BERGEN(n) = 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42, -17/2, 127/30, 155/2,..
Difference table:
1,       1/2,     -1/6,     -1/2,      7/30,        3/2,    -31/42,...
-1/2,   -2/3,     -1/3,    11/15,     19/15,     -47/21,   -163/21,...
-1/6,    1/3,    16/15,     8/15,  -368/105,    -116/21,  2152/105,...
1/2,   11/15,    -8/15, -424/105,  -212/105,   2732/105,  4204/105,...
7/30, -19/15, -368/195,  212/105,  2944/105,   1472/105,...
-3/2, -47/21,   116/21, 2732/105, -1472/105, -70240/231, -35120/231,... .
a(n) is an autosequence. Its inverse binomial transform is the sequence signed. Its main diagonal is the double of the first upper diagonal.
a(n) is divisible by A051716(n+1).
Denominators of the main diagonal: A181131(n). Checked by Jean-François Alcover for the first 25 terms.
The numerators of the main diagonal:
1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272,...
(thanks to Jean-François Alcover) are divisible by 2^n.

Cf. A083420.

A225825 := proc(n)
    local nhalf ;
    nhalf := floor(n/2) ;
    if type(n,'even') then
        A001896(nhalf) ;
    else
        (-1)^nhalf*A110501(nhalf+1) ;
    end if;
end proc; # R. J. Mathar, Oct 28 2013

a[0] = 1; a[n_] := Numerator[BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 01 2013 *)

c(n)=(0 followed by -A036968(n+1)) = 0, 1, 0, -1, 0, 3,... .

a(n) = A157779(n) + c(n).

More terms from Jean-François Alcover, Aug 01 2013

Definition corrected by R. J. Mathar, Oct 28 2013

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).

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.

A335947 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 7, 0, -1, 0, 1, 0, 7, 0, -5, 0, 1, -31, 0, 7, 0, -5, 0, 1, 0, -31, 0, 49, 0, -7, 0, 1, 127, 0, -31, 0, 49, 0, -7, 0, 1, 0, 381, 0, -31, 0, 147, 0, -3, 0, 1, -2555, 0, 381, 0, -155, 0, 49, 0, -15, 0, 1

Offset: 0

Peter Luschny, Jul 01 2020

The polynomials form an Appell sequence.

The parity of n equals the parity of b(n, x). The Bernoulli polynomials do not possess this property.

			First few polynomials are:
b_0(x) = 1;
b_1(x) = x;
b_2(x) = -(1/12) + x^2;
b_3(x) = -(1/4)*x + x^3;
b_4(x) = (7/240) - (1/2)*x^2 + x^4;
b_5(x) = (7/48)*x - (5/6)*x^3 + x^5;
b_6(x) = -(31/1344) + (7/16)*x^2 - (5/4)*x^4 + x^6;
Normalized by A335949:
b_0(x) = 1;
b_1(x) = x;
b_2(x) = (-1 + 12*x^2) / 12;
b_3(x) = (-x + 4*x^3) / 4;
b_4(x) = (7 - 120*x^2 + 240*x^4) / 240;
b_5(x) = (7*x - 40*x^3 + 48*x^5) / 48;
b_6(x) = (-31 + 588*x^2 - 1680*x^4 + 1344*x^6) / 1344;
b_7(x) = (-31*x + 196*x^3 - 336*x^5 + 192*x^7) / 192;
Triangle starts:
[0] 1;
[1] 0,   1;
[2] -1,  0,   1;
[3] 0,   -1,  0,   1;
[4] 7,   0,   -1,  0,   1;
[5] 0,   7,   0,   -5,  0,  1;
[6] -31, 0,   7,   0,   -5, 0,   1;
[7] 0,   -31, 0,   49,  0,  -7,  0,  1;
[8] 127, 0,   -31, 0,   49, 0,   -7, 0,  1;
[9] 0,   381, 0,   -31, 0,  147, 0,  -3, 0, 1;

Cf. A335948 (denominators), A335949 (denominators of the polynomials).
Cf. A157779 (column 0),  A001896 (column 0 at even indices only).
Cf. A164555/A027642, A157781/A157782, A157779/A157780, A196838/A196839.

b := (n,x) -> bernoulli(n, x+1/2):
A335947row := n -> seq(numer(coeff(b(n,x), x, k)), k = 0..n):
seq(A335947row(n), n = 0..10);

b(n, 1/2) = Bernoulli(n, 1) = A164555(n)/A027642(n).
b(n, -1) = Bernoulli(n, -1/2) = A157781(n)/A157782(n).
b(n, 0) = Bernoulli(n, 1/2) = A157779(n)/A157780(n).
b(n, x) = Bernoulli(n, x + 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.

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).

A326582 A signed variant of A309132.

Original entry on oeis.org

1, 1, 1, 4, 1, 9, 1, 16, 27, 25, 1, 36, 1, 49, -75, 64, 1, 81, -1, 100, 49, 121, -1, 144, 125, 169, -243, 196, 1, 225, -1, 256, 363, 289, -1225, 324, 1, 361, -169, 400, 1, 441, -1, 484, 135, 529, -1, 576, 343, 625, -867, 676, 1, 729, -3025, 784, 361, 841, -1

Offset: 0

Peter Luschny, Jul 15 2019

sign

See A309132 for background and conjectures.

Cf. A309132, A157779, A157780.

nB := n -> numer(bernoulli(n-1,1/2)): dB := n -> denom(bernoulli(n-1,1/2)):
R := n -> n/(nB(n) + dB(n)/n): a := n -> numer(R(n+1)/4^irem(n,2)):
seq(a(n), n=0..58);

a(n) = numerator(R(n+1)/4^mod(n,2)) with R(n) = n/(nB(n) + dB(n)/n) and nB(n) = numerator(B(n-1, 1/2)), dB(n) = denominator(B(n-1, 1/2)) where B(n, x) denotes the Bernoulli polynomials.
|a(2*n)| = A309132(2*n + 1) for n >= 0.
a(2*n+1) = (n + 1)^2 for n >= 0.

cp's OEIS Frontend

A141459 a(n) = Product_{p-1 divides n} p, where p is an odd prime.

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A157780 Denominator of Bernoulli(n, 1/2).

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A224783 Denominator of Bernoulli(n,1/2) - Bernoulli(n,0).

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A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

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

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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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A335947 T(n, k) = numerator([x^k] b_n(x)), where b_n(x) = Sum_{k=0..n} binomial(n,k)* Bernoulli(k, 1/2)*x^(n-k). Triangle read by rows, for n >= 0 and 0 <= k <= n.

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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).

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.