A176591 -id:A176591 - cp's OEIS frontend

A176328 Numerators of the rational sequence with e.g.f. (x/2)*(exp(-x) + 1)/(exp(x) - 1).

1, -1, 7, -3, 59, -5, 127, -7, 119, -9, 335, -11, 15689, -13, 49, -15, 463, -17, 51049, -19, -171311, -21, 856031, -23, -236331331, -25, 8553181, -27, -23749448849, -29, 8615841490835, -31, -7709321033057, -33, 2577687858469

Offset: 0

Paul Curtz, Apr 15 2010

Numerator of the n-term of the inverse binomial transform of the modified Bernoulli sequence A176327(k)/A027642(k).
The sequence of modified Bernoulli numbers A176327(k)/A027642(k) is defined to be the same as the Bernoulli sequence, except the term at index k=1 which is zero.
Its inverse binomial transform is 1, -1, 7/6, -3/2, 59/30, -5/2, 127/42, -7/2, 119/30, -9/2, 335/66, -11/2, ...; the numerators define this sequence here.

			The first few of the polynomials mentioned in the formula section are: 1, 1/2, 1/6 + x^2, (3/2)*x^2, -1/30 + x^2 + x^4, (5/2)*x^4, 1/42 - (1/2)*x^2 +(5/2)*x^4 + x^6, (7/2)*x^6, -1/30 + (2/3)*x^2 - (7/3)*x^4 + (14/3)*x^6 + x^8, (9/2)*x^8, ...  The values of these polynomials at x=1 start 1, 1/2, 7/6, 3/2, 59/30, 5/2, 127/42, 7/2, ... - _Peter Luschny_, Aug 18 2018

Cf. A176591 (denominators), A141056 (denominators for the unsigned variant).

read("transforms") ; evb := [1,0,seq(bernoulli(n),n=2..50)] ; BINOMIALi(evb) ; apply(numer,%) ; # R. J. Mathar, Dec 01 2010
seq(numer((-1)^n*(bernoulli(n,1)+bernoulli(n,2))/2),n=0..34); # Peter Luschny, Jun 17 2012
gf := cosh(x*z)*z/(1-exp(-z)): ser := series(gf, z, 35):
seq((-1)^n*numer(subs(x=1, n!*coeff(ser, z, n))), n=0..34); # Peter Luschny, Aug 19 2018

terms = 35; egf = (x/2)*((Exp[-x] + 1)/(Exp[x] - 1)) + O[x]^(terms);
CoefficientList[egf, x]*Range[0, terms-1]! // Numerator (* Jean-François Alcover, Jun 13 2017 *)

my(x = 'x + O('x^50)); apply(x->numerator(x), Vec(serlaplace((x/2)*(exp(-x) + 1)/(exp(x) - 1)))) \\ Michel Marcus, Aug 19 2018

Conjecture: a(2*n+1) = -2*n-1.
a(n) = numerator((-1)^n*(bernoulli(n, 1) + bernoulli(n, 2))/2). - Peter Luschny, Jun 17 2012
(-1)^n*a(n) are the numerators of the polynomials generated by cosh(x*z)*z/(1-exp(-z)) evaluated x=1 (see the example section). The denominators of these values are A141056. - Peter Luschny, Aug 18 2018

Apparently incorrect claims concerning the inverse binomial transform of the B_n removed by R. J. Mathar, Dec 01 2010

New name from Peter Luschny, Jun 17 2012

A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

Original entry on oeis.org

0, 1, 1, 7, 8, 11, 10, 7, 8, 19, 14, 337, 1028, 5, -2, -1681, 1936, 22133, -21734, -87223, 87388, 427291, -427222, -118181363, 118182728, 4276553, -4276550, -11874730297, 11874730732, 4307920641583

Offset: 0

Paul Curtz, Sep 13 2011

The array of (-1)^n*A176328(n)/A176591(n) and its first, second, etc. differences in subsequence rows starts as follows:
0,     1,     2,   19/6,   14/3, 199/30,    137/15, ... (-1)^n * A176328(n)/A176591(n),
1,     1,   7/6,    3/2,  59/30,    5/2,    127/42, ... see A176328,
0,   1/6,   1/3,   7/15,   8/15,  11/21,     10/21, ...
1/6, 1/6,  2/15,   1/15, -1/105,  -1/21,    -1/105, ... see A190339
0, -1/30, -1/15, -8/105, -4/105,  4/105, -116/1155, ...
The numerators in the 3rd row, 0, 1/6, 1/3, 7/15, 8/15, 11/21, 10/21, 7/15, 8/15, 19/33, 14/33, 337/1365, 1028/1365, 5/3, -2/3, -1681/255, 1936/255, ... define the current sequence.
The associated denominators are 1, 6 and followed by 3, 15, 15 etc as provided in A172087.
The second column of the array, 1, 1, 1/6, 1/6, -1/30, -1/30, ... contains doubled A000367(n)/A002445(n). These are related to A176150, A176144, and A176184.
In the first subdiagonal of the array we see 1, 1/6, 2/15, -8/150, 8/105, -32/321, 6112/15015, -3712/2145 , ... continued as given by A181130 and A181131.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

read("transforms") ;
evb := [0, 1, 0, seq(bernoulli(n), n=2..30)] ;
ievb := BINOMIALi(evb) ;
[seq((-1)^n*op(n,ievb),n=1..nops(ievb))] ;
DIFF(%) ;
DIFF(%) ;
apply(numer,%) ; # R. J. Mathar, Sep 20 2011

evb = Join[{0, 1, 0}, Table[BernoulliB[n], {n, 2, 32}]]; ievb = Table[ Sum[Binomial[n, k]*evb[[k+1]], {k, 0, n}], {n, 0, Length[evb]-3}]; Differences[ievb, 2] // Numerator (* Jean-François Alcover, Sep 09 2013, after R. J. Mathar *)

a(2*n+1) + a(2*n+2) = A172087(2*n+2) = A172087(2*n+3), n >= 1.

A176618 The numerator of the n-th term of the inverse binomial transform of the sequence 0, 1, 0, B_2, B_3, B_4, .. of modified Bernoulli numbers.

Original entry on oeis.org

0, 1, -2, 19, -14, 199, -137, 851, -548, 4121, -2533, 67451, -40078, 404869, -234967, 1655047, -940136, 32428087, -18383711, 439693871, -235204778, -724823111, 352226881, 260572074487, -130542594044, -6002444699183, 3000757572779

Offset: 0

Paul Curtz, Apr 22 2010

The starting sequence contains the terms A176327(.)/A176591(.) prefixed with a single zero (which occupies the term at index zero), basically 0, 1, 0 followed by the Bernoulli numbers without B_0 and B_1.
Its inverse binomial transform is 0, 1, -2, 19/6, -14/3, 199/30, -137/15, 851/70, -548/35, 4121/210, -2533/105, 67451/2310, -40078/1155, 404869/10010, -234967/5005, 1655047/30030,.. and taking numerators defines the current sequence.
The denominators of the transformed sequence appear to be A172031, checked up to A176618(33).

read("transforms") ;
evb := [0, 1, 0, seq(bernoulli(n), n=2..50)] ;
ievb := BINOMIALi(evb) ;
apply(numer,%) ;

A228767 Second bisection of the inverse binomial transform of the rational sequence with e.g.f. (x/2)*(exp(-x)+1)/(exp(x)-1).

Original entry on oeis.org

-2, -9, -45, -231, -1161, -5643, -26637, -122895, -557073, -2490387, -11010069, -48234519, -209715225, -905969691, -3892314141, -16642998303, -70866960417, -300647710755, -1271310319653, -5360119185447, -22539988369449, -94557999988779, -395824185999405

Offset: 1

Michel Marcus, following a suggestion of Paul Curtz, Sep 03 2013

sign

The sequence to be transformed is A176328/A176591, its inverse binomial transform begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, -5643, 33578309/2730, ...
Its first bisection is constituted of fractional numbers, with denominators A176591, whereas this bisection is constituted of integers only.
It appears that a(1) = -2 and a(n) = -1 * A005408(n-1) * A087289(n-2) for n>1.

fr(n) = if (n==0, 1, (-1)^n*(subst(bernpol(n), x, 1) + subst(bernpol(n), x, 2))/2);
ibtfr(n) = sum(k = 0, n, (-1)^(n-k)*binomial(n, k) * fr(k));
lista(nn) = {forstep(n=1, nn, 2, print1(ibtfr(n), ", "););} \\ Michel Marcus, Sep 03 2013

Conjecture: G.f. -x*(2-11*x+21*x^2-2*x^3+8*x^4)/((1-x)^2*(1-4*x)^2). [Bruno Berselli, Sep 03 2013]

Conjecture: a(n) = (8+4^n)*(1-2*n)/8 for n>1, a(1)=-2. [Bruno Berselli, Sep 03 2013]

A176447 a(2n) = -n, a(2n+1) = 2n+1.

Original entry on oeis.org

0, 1, -1, 3, -2, 5, -3, 7, -4, 9, -5, 11, -6, 13, -7, 15, -8, 17, -9, 19, -10, 21, -11, 23, -12, 25, -13, 27, -14, 29, -15, 31, -16, 33, -17, 35, -18, 37, -19, 39, -20, 41, -21, 43, -22, 45, -23, 47, -24, 49, -25, 51, -26, 53, -27, 55, -28, 57, -29, 59, -30, 61, -31, 63, -32, 65, -33, 67, -34, 69, -35

Offset: 0

Paul Curtz, Apr 18 2010

There is more complicated way of defining the sequence: consider the sequence of modified Bernoulli numbers EVB(n) = A176327(n)/A176289(n) and its inverse binomial transform IEVB(n) = A176328(n)/A176591(n). Then a(n) is the numerator of the difference EVB(n)-IEVB(n). The denominator of the difference is 1 if n=0, else A040001(n-1).
A particularity of EVB(n) is: its (forward) binomial transform is 1, 1, 7/6, 3/2, 59/30,.. = (-1)^n*IEVB(n).
Note that A026741 is related to the Rydberg-Ritz spectrum of the hydrogen atom.

			G.f. = x - x^2 + 3*x^3 - 2*x^4 + 5*x^5 - 3*x^6 + 7*x^7 - 4*x^8 + 9*x^9 - 5*x^10 + ...

R. J. Mathar, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008795, A026741, A040001, A176289, A176327, A176328, A176591.

[n*(1-3*(-1)^n)/4: n in [0..60]]; // Vincenzo Librandi, Aug 04 2011

a[n_?EvenQ]:=-(n/2); a[n_?OddQ]:=n; Table[a[n], {n, 100}] (* Alonso del Arte, Dec 01 2010 *)
a[ n_] := n / If[ Mod[ n, 2] == 1, 1, -2]; (* Michael Somos, Jun 11 2013 *)
CoefficientList[Series[x (1 - x + x^2)/((x - 1)^2*(1 + x)^2), {x, 0, 70}], x]  (* Michael De Vlieger, Dec 10 2016 *)
LinearRecurrence[{0,2,0,-1},{0,1,-1,3},80] (* Harvey P. Dale, Nov 01 2017 *)

{a(n) = n / if( n%2, 1, -2)}; /* Michael Somos, Jun 11 2013 */

From R. J. Mathar, Dec 01 2010:  (Start)
a(n) = (-1)^n*A026741(n) = n*(1-3*(-1)^n)/4.
G.f.: x*(1-x+x^2) / ( (x-1)^2*(1+x)^2 ).
a(n) = +2*a(n-2) -a(n-4).  (End)
a(n) = -a(-n) for all n in Z. - Michael Somos, Jun 11 2013
From Michael Somos, Aug 30 2014: (Start)
Euler transform of length 6 sequence [ -1, 3, 1, 0, 0, -1].
0 = - 1 - a(n) - a(n+1) + a(n+2) + a(n+3) for all n in Z.
0 = 1 + a(n)*(-2 -a(n) + a(n+2)) - 2*a(n+1) - a(n+2) for all n in Z. (End)
From Michael Somos, May 04 2015: (Start)
a(n) is multiplicative with a(2^e) = -(2^(e-1)) if e>0, a(p^e) = p^e otherwise.
G.f.: (f(x) - 3 * f(-x)) / 4 where f(x) := x / (1 - x)^2.
G.f.: x * (1 - x) * (1 - x^6) / ((1 - x^2)^3 * (1 - x^3)). (End)
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: zeta(s-1) * (1 - 3/2^s).
Sum_{k=0..n} a(k) = A008795(n-1), for n > 0.
Sum_{k=0..n} a(k) ~ n^2/8. (End)

A228827 Numerators of the first bisection of the inverse binomial transform of the rational sequence with e.g.f. (x/2)*(exp(-x)+1)/(exp(x)-1).

Original entry on oeis.org

1, 25, 599, 4285, 15599, 169625, 33578309, 344155, 133697983, 941417335, 1729982389, 3184334285, 274574499509, 2625798955, 1611022490371, 123951819730625, 9814145542783, 3453861186955, -25128299959971711973, 2945661954537595, -260933954573210488051

Offset: 0

Paul Curtz & Michel Marcus, Sep 06 2013

The sequence to be transformed is A176328/A176591, its inverse binomial transform begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, -5643, 33578309/2730, ...

It appears that a(n) - A000367(n) is a multiple of A002445(n), and the quotients are 0, 4, 20, 102, 520, 2570, 12300, ...

Cf. A228767 (other bisection).

fr(n) = {default(seriesprecision, n+1); egf = (x/2)*(exp(-x)+1)/(exp(x)-1);(n)!* polcoeff(egf, n);}
ibtfr(n) = sum(k = 0, n, (-1)^(n-k)*binomial(n, k) * fr(k));
lista(nn) = {forstep(n = 0, nn, 2, print1(numerator(ibtfr(n)), ", "););} \\ Michel Marcus, Sep 06 2013

A227978 a(0)=1, a(1)=2; for n>1, a(n) = n*(2^n+4)/4.

Original entry on oeis.org

1, 2, 4, 9, 20, 45, 102, 231, 520, 1161, 2570, 5643, 12300, 26637, 57358, 122895, 262160, 557073, 1179666, 2490387, 5242900, 11010069, 23068694, 48234519, 100663320, 209715225, 436207642, 905969691, 1879048220, 3892314141, 8053063710, 16642998303

Offset: 0

Paul Curtz, Oct 07 2013

The inverse binomial transform of A176328/A176591 (see Comments field in A228827) begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, ... Consider these values without sign and the fractions rounded to the nearest integer, the sequence lists the resulting numbers.
Differences table of a(n):
  1, 2,  4,  9, 20,  45, 102, 231,  520, 1161, ...
  1, 2,  5, 11, 25,  57, 129, 289,  641, 1409, ... After 2: 2^m*(m+4)+1.
  1, 3,  6, 14, 32,  72, 160, 352,  768, 1664, ... A078836 (after 3).
  2, 3,  8, 18, 40,  88, 192, 416,  896, 1920, ... A129955.
  1, 5, 10, 22, 48, 104, 224, 480, 1024, 2176, ... A079861 (after 5).
  4, 5, 12, 26, 56, 120, 256, 544, 1152, 2432, ... After 5: 2^m*(m+12).
  1, 7, 14, 30, 64, 136, 288, 608, 1280, 2688, ... After 7: 2^m*(m+14).
  6, 7, 16, 34, 72, 152, 320, 672, 1408, 2944, ..., etc.
(n-1)*a(n)-n*a(n-1) = A001788(n-1) for n>1. [Bruno Berselli, Oct 11 2013]

Bruno Berselli, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A001788, A079862, A079863.

[1,2] cat [n*(2^n+4)/4: n in [2..40]]; // Bruno Berselli, Oct 11 2013

Join[{1, 2}, Table[n (2^n + 4)/4, {n, 2, 40}]] (* Bruno Berselli, Oct 11 2013 *)

a(n) = if (n == 0, 1, if (n == 1, 2, n*(2^n+4)/4)); \\ Michel Marcus, Oct 11 2013

a(2n+2) = A229135(n+1); a(2n-1) = -A228767(n) for n>0.
a(n) = 6*a(n-1) -13*a(n-2) +12*a(n-3) -4*a(n-4) for n>5.
G.f.: (1-4*x+5*x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1-2*x)^2). - Colin Barker, Oct 09 2013

More terms from Colin Barker, Oct 09 2013

A249306 Denominators A027642(n) of Bernoulli numbers except for a(4*k+5)=2 instead of 1.

Original entry on oeis.org

1, 2, 6, 1, 30, 2, 42, 1, 30, 2, 66, 1, 2730, 2, 6, 1, 510, 2, 798, 1, 330, 2, 138, 1, 2730, 2, 6, 1, 870, 2, 14322, 1, 510, 2, 6, 1, 1919190, 2, 6, 1, 13530, 2, 1806, 1, 690, 2, 282, 1, 46410, 2, 66, 1, 1590, 2, 798, 1, 870, 2, 354, 1

Offset: 0

Paul Curtz, Oct 28 2014

nonn

There exist an infinity of 1's, 2's, 6's, 30's, 42's, 66's, ... .
Respective ranks:
   0,   3,    7,   11,   15,   19, ...
   1,   5,    9,   13,   17,   21, ...   (= A016813)
   2,  14,   26,   34,   38,   62, ...   (= A051222)
   4,   8,   68,   76,  124,  152, ...   (= A051226)
   6, 114,  186,  258,  354,  402, ...   (= A051228)
  10,  50,  170,  370,  470,  590, ...   (= A051230)
  12,  24, 1308, 1884, 2004, 2364, ...   (= A249134)
  etc.
Hence by antidiagonals a permutation of A001477(n).
First column: A248614(n).
a(n) is an alternative sequence for the denominators of the Bernoulli numbers.
First 36 terms of the corresponding clockwise spiral:
.
  330------2----138------1---2730------2
    |                                  |
    |                                  |
    1     42------1-----30------2      6
    |      |                    |      |
    |      |                    |      |
  798      2      1------2     66      1
    |      |             |      |      |
    |      |             |      |      |
    2     30------1------6      1    870
    |                           |      |
    |                           |      |
  510------1------6------2---2730      2
                                       |
                                       |
    1------6------2----510------1--14322

Index entries for sequences related to Bernoulli numbers

A variant of the Clausen numbers A141056, A160014. And of A176591.

Cf. A000034, A002445, A016813, A027642, A051222, A051226, A051228, A051230, A090126, A164020, A248614, A249134.

Clausen := proc(n) local S, i;
S := numtheory[divisors](n); S := map(i->i+1, S);
S := select(isprime, S); mul(i, i=S) end:
A249306 := n -> `if`(n mod 4 = 3, 1, Clausen(n)):
seq(A249306(n), n=0..59); # Peter Luschny, Nov 10 2014

a[n_] := Denominator[BernoulliB[n]]; a[n_ /; Mod[n, 4] == 1] = 2; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2014 *)

a(2n) = A002445(n), a(2n+1) = A000034(n+1).

A257106 Denominators of the inverse binomial transform of the Bernoulli numbers with B(1)=2/3.

Original entry on oeis.org

1, 3, 6, 2, 10, 6, 42, 6, 30, 2, 22, 6, 2730, 6, 6, 2, 170, 6, 798, 6, 330, 2, 46, 6, 2730, 6, 6, 2, 290, 6, 14322, 6, 510, 2, 2, 6, 1919190, 6, 6, 2, 4510, 6, 1806, 6, 690, 2, 94, 6, 46410, 6, 66, 2, 530, 6, 798, 6, 870, 2, 118, 6, 56786730, 6, 6, 2, 170, 6

Offset: 0

Paul Curtz, Apr 23 2015

nonn

Difference table of Bernoulli numbers with B(1)=2/3:
1,       2/3,    1/6,      0, -1/30,     0,  1/42, 0, ...
-1/3,   -1/2,   -1/6,  -1/30,  1/30,  1/42, -1/42, ...
-1/6,    1/3,   2/15,   1/15, -1/105, -1/21, ...
1/2,    -1/5,  -1/15, -8/105, -4/105, ...
-7/10,  2/15, -1/105,  4/105, ...
5/6,    -1/7,   1/21, ...
-41/42, 2/15, ...
7/6, ...
...
First column: 1, -1/3, -1/6, 1/2, -7/10, 5/6, -41/42, 7/6, -41/30, 3/2, -35/22, 11/6, ... . a(n) is the n-th term of the denominators.
Antidiagonal sums: 1, 1/3, -1/2, 2/3, -5/6, 1, -7/6, 4/3, -3/2, 5/3, -11/6, 2, ... . See A060789(n).
a(2n+2)/a(2n+1) = 2, 5, 7, 5, 11, 455, ... .
By definition, for B(1) = b, the inverse binomial transform is
Bi(b) =       1,  -1 + b, 7/6 - 2*b, -3/2 + 3*b, 59/30 + 4*b, ...
      = A176328(n)/A176591(n) - (-1)^n *n*b.
With Bic(b) = 0, -1/2 + b, 1 - 2*b,  -3/2 + 3*b,   2 + 4*b, ...
            = (-1)^n *(A001477(n)/2 - n*b),
Bi(b) = (-1)^n *(A164555(n)/A027642(n) + A001477(n)/2 - n*b) =
      = A027641(n)/A027642(n) + Bic(b) .

			a(0) = 1-0, a(1) = -1/2 +1/6 = -1/3, a(2) = 1/6 -1/3 = -1/6, a(3) = 0 +1/2.

Cf. A256595, A027641/A027642(n), A164555(n)/A027642(n), A060789, A176328/A176591, A001477, A109007.

max = 66; B[1] = 2/3; B[n_] := BernoulliB[n]; BB = Array[B, max, 0]; a[n_] := Differences[BB, n] // First // Denominator; Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, May 11 2015 *)

def A257106_list(len, B1) :
    T = matrix(QQ, 2*len+1)
    for m in (0..2*len) :
        T[0, m] = bernoulli_polynomial(1, m) if m <> 1 else B1
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
    return [denominator(T[k, 0]) for k in (0..len-1)]
A257106_list(66, 2/3) # Peter Luschny, May 09 2015

Conjecture: a(2n+1) = 3 followed by period 3: repeat 2, 6, 6.
Conjecture: a(2n) = A002445(n)/(period 3: repeat 1, 1, 3).
a(n) = A027641(n)/A027642(n) - (-1)^n *n/6.

A257935 Numerators of the inverse binomial transform of the Bernoulli numbers with B(1)=1.

Original entry on oeis.org

1, 0, -5, 3, -61, 5, -125, 7, -121, 9, -325, 11, -17071, 13, -35, 15, -7697, 17, 36685, 19, -177911, 21, 852995, 23, -236396851, 25, 8553025, 27, -23749473209, 29, 8615841061175, 31, -7709321049377, 33, 2577687858265, 35, -26315271553088022793, 37

Offset: 0

Paul Curtz, May 13 2015

sign

Difference table of 1, 1, 1/6, 0, -1/30, ... :
1,            1,    1/6,      0,  -1/30,     0,  1/42, 0, ...
0,         -5/6,   -1/6,  -1/30,   1/30,  1/42, -1/42, ...
-5/6,       2/3,   2/15,   1/15, -1/105, -1/21, ...
3/2,      -8/15,  -1/15, -8/105, -4/105, ...
-61/30,    7/15, -1/105,  4/105, ...
5/2,     -10/21,   1/21, ...
-125/42,  11/21, ...
7/2, ...
etc.
The inverse binomial transform is the first column. a(n) is the n-th term of the numerators. See A027641(n+1).
Denominators: A176591.
Is a(4n+2) a multiple of 5? This is true, at least up to 4n+2 = 998. - Jean-François Alcover, Jul 02 2015

			By the first formula: numerators of 1-0=1, -1/2+1/2=0, 1/6-1=-5/6, 0+3/2=3/2,....

Colin Barker, Table of n, a(n) for n = 0..629

Cf. A257106, A027641/A027642, A164555/A027642, A176327/A027642, A176328/A176591, A026741.

max = 40; B[1] = 1; B[n_] := BernoulliB[n]; BB = Array[B, max, 0]; a[n_] := Differences[BB, n] // First // Numerator; Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, May 20 2015 *)

firstdiff(s) = my(t=vector(#s-1)); for(i=2, #s, t[i-1]=s[i]-s[i-1]); t
a257935(k) = {
  my(s=[], b = concat([1,1], vector(k, n, n++; bernfrac(n))));
  until(#b<2,
    s = concat(s, numerator(b[1]));
    b = firstdiff(b)
  );
  s
}
a257935(50) \\ Colin Barker, May 13 2015

a(n) = numerators of A027641(n)/A027642(n) - (-1)^n*n/2.
a(n) = (A176328(n) - (-1)^n*n)*A176591(n).
a(n) = 2*A027641(n)*A176591(n)/A027642(n) - A176328(n).

More terms from Colin Barker, May 13 2015

cp's OEIS Frontend

A176328 Numerators of the rational sequence with e.g.f. (x/2)*(exp(-x) + 1)/(exp(x) - 1).

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A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

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A176618 The numerator of the n-th term of the inverse binomial transform of the sequence 0, 1, 0, B_2, B_3, B_4, .. of modified Bernoulli numbers.

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A228767 Second bisection of the inverse binomial transform of the rational sequence with e.g.f. (x/2)*(exp(-x)+1)/(exp(x)-1).

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A176447 a(2n) = -n, a(2n+1) = 2n+1.

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A228827 Numerators of the first bisection of the inverse binomial transform of the rational sequence with e.g.f. (x/2)*(exp(-x)+1)/(exp(x)-1).

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A227978 a(0)=1, a(1)=2; for n>1, a(n) = n*(2^n+4)/4.

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A249306 Denominators A027642(n) of Bernoulli numbers except for a(4*k+5)=2 instead of 1.

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