A129826 -id:A129826 - cp's OEIS frontend

A051717 1, followed by denominators of first differences of Bernoulli numbers (B(i)-B(i-1)).

1, 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, 2730, 6, 6, 510, 510, 798, 798, 330, 330, 138, 138, 2730, 2730, 6, 6, 870, 870, 14322, 14322, 510, 510, 6, 6, 1919190, 1919190, 6, 6, 13530, 13530, 1806, 1806, 690, 690, 282, 282, 46410, 46410, 66, 66, 1590, 1590

Offset: 0

N. J. A. Sloane

Equivalently, denominators of Bernoulli twin numbers C(n) (cf. A051716).
The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to Paul Curtz.
Denominators of column 1 of table described in A051714/A051715.

			Bernoulli numbers: 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...
First differences: -3/2, 2/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, ...
Numerators: -3, 2, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, ...
Denominators: 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, ...
Sequence of C(n)'s begins: 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A027641, A027642, A051714, A051715, A051716, A129825, A129826.
Cf. A129724.
For numerators see A172083.

f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
function A051717(n)
  if n eq 0 then return 1;
  elif (n mod 2) eq 0 then return Denominator(f(n));
  else return Denominator(-f(n));
  end if;
end function;
[A051717(n): n in [0..50]]; // G. C. Greubel, Apr 22 2023

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;

c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Denominator[c[n]], {n,0,53}] (* Jean-François Alcover, Dec 19 2011 *)
Join[{1},Denominator[Total/@Partition[BernoulliB[Range[0,60]],2,1]]] (* Harvey P. Dale, Mar 09 2013 *)
Join[{1},Denominator[Differences[BernoulliB[Range[0,60]]]]] (* Harvey P. Dale, Jun 28 2021 *)

a(n)=if(n<3,n+1,denominator(bernfrac(n)-bernfrac(n-1))) \\ Charles R Greathouse IV, May 18 2015

def f(n): return bernoulli(n)+bernoulli(n-1)
def A051717(n):
    if (n==0): return 1
    elif (n%2==0): return denominator(f(n))
    else: return denominator(-f(n))
[A051717(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

More terms from James Sellers, Dec 08 1999
Edited by N. J. A. Sloane, May 25 2008
Entry revised by N. J. A. Sloane, Apr 22 2021

A051716 Numerators of Bernoulli twin numbers C(n).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, -7, -3617, 3617, 43867, -43867, -174611, 174611, 854513, -854513, -236364091, 236364091, 8553103, -8553103, -23749461029, 23749461029, 8615841276005, -8615841276005, -7709321041217, 7709321041217, 2577687858367

Offset: 0

N. J. A. Sloane

The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to Paul Curtz.
For denominators see A051717.
Negatives of numerators of column 1 of table described in A051714/A051715.

			The C(n) sequence is 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..640
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Cf. A000367, A027641, A027642, A051714, A051715.

Cf. A051717, A129825, A129826, A129724, A164555.

f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
function A051716(n)
  if n eq 0 then return 1;
  elif (n mod 2) eq 0 then return Numerator(f(n));
  else return Numerator(-f(n));
  end if;
end function;
[A051716(n): n in [0..50]]; // G. C. Greubel, Apr 22 2023

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;

c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Numerator[c[n]], {n,0,34}] (* Jean-François Alcover, Dec 19 2011 *)

a(n) = if (n==0, 1, nu = numerator(bernfrac(n)+bernfrac(n-1)); if (n%2, -nu, nu)); \\ Michel Marcus, Jan 29 2017

def f(n): return bernoulli(n)+bernoulli(n-1)
def A051716(n):
    if (n==0): return 1
    elif (n%2==0): return numerator(f(n))
    else: return numerator(-f(n))
[A051716(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

Numerators of differences of the sequence of rational numbers 0 followed by A164555/A027642. - Paul Curtz, Jan 29 2017

The e.g.f. of the rationals a(n)/A051717(n) is -(1/x + x^2/2 + x/(1 - exp(x)) + dilog(exp(-x))), (with dilog(x) = polylog(2, 1-x)). From integrating the e.g.f. of the z-sequence (exp(x) - (1+x))/(exp(x) -1)^2 for the Bernoulli polynomials of the second kind (A290317 / A290318). - Wolfdieter Lang, Aug 07 2017

More terms from James Sellers, Dec 08 1999

Edited by N. J. A. Sloane, May 25 2008

A129825 a(n) = n!*Bernoulli(n-1), n > 2; a(0)=0, a(1)=1, a(2)=1.

Original entry on oeis.org

0, 1, 1, 1, 0, -4, 0, 120, 0, -12096, 0, 3024000, 0, -1576143360, 0, 1525620096000, 0, -2522591034163200, 0, 6686974460694528000, 0, -27033456071346536448000, 0, 160078872315904478576640000, 0, -1342964491649083924630732800000, 0, 15522270327163593186886877184000000, 0

Offset: 0

Paul Curtz, Jun 03 2007

sign

Define "conjugated" Bernoulli numbers G(n) via G(0)=0, G(1)=B(0)=1, G(2)=-B(1)=1/2, G(n+1)=B(n), where B(n)=A027641(n)/A027642(n).
The sequence is then defined by a(n) = n!*G(n).
The first differences are 1, 0, 0, -1, -4, 4, 120, -120, -12096, ...
The 2nd differences are -1, 0, -1, -3, 8, 116, -240, -11976, 24192, 3011904, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A001332, A027641, A027642, A129716, A129814, A129826.
Equals second left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A161742 and A161743.
Cf. A094310 [T(n,k) = n!/k], A008277 [S2(n,k); Stirling numbers of the second kind], A028246 [Worpitzky's triangle] and A008955 [CFN triangle].

[n le 2 select Floor((n+1)/2) else Factorial(n)*Bernoulli(n-1): n in [0..40]]; // G. C. Greubel, Apr 26 2024

A129825 := proc(n) if n <= 1 then n; elif n = 2 then 1; else n!*bernoulli(n-1) ; fi; end: # R. J. Mathar, May 21 2009

a[n_] := n!*BernoulliB[n-1]; a[0]=0; a[2]=1; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Mar 04 2013 *)

[(n+1)//2 if n <3 else factorial(n)*bernoulli(n-1) for n in range(41)] # G. C. Greubel, Apr 26 2024

From Johannes W. Meijer, Jun 18 2009: (Start)
a(n) = Sum_{k=1..n} (-1)^(k+1)*(n!/k)*S2(n, k)*(k-1)!.
a(n) = Sum_{k=0..n-1} ((-1)^k/(k!*(k+1)!))*n!*A028246(n, k+1) *A008955(k, k). (End)
a(n) = A129814(n-1) for n > 2. - Georg Fischer, Oct 07 2018

Edited by R. J. Mathar, May 21 2009

A129378 Row sums of coefficients of Bernoulli twin number polynomials.

Original entry on oeis.org

1, 1, 4, 20, 116, 744, 5160, 39360, 350784, 3749760, 42940800, 442713600, 4650877440, 109244298240, 2833294464000, -3487131648000, -2166903606067200, 51809012320665600, 6808619561103360000, -131306587205713920000, -26982365129174827008000, 595860034297401409536000

Offset: 0

Paul Curtz, Jun 08 2007

sign

The origin of the sequence are polynomials on pages 61 and 69 of the CCSA paper. The first few of the polynomials have been noted in the 1992 Gazette paper.
We construct Bernoulli twin numbers polynomials C(n,x) = Sum_{j=1..n} binomial(n-1,j-1)*B(j,x) where B(n,x) are the Bernoulli polynomials of A048998 and A048999 and where binomial(.,.) is the Pascal triangle A007318: C(0,x)=B(0,x); C(1,x)=B(1,x); C(2,x)=B(2,x)+B(1,x); C(3,x)=B(3,x)+2B(2,x)+B(1,x).
The triangle of coefficients [x^m] C(n,x) for rows n=0,1,2,.. and decreasing power m=n,...,0 along each row starts
  1;
  1, -1/2;
  1,    0, -1/3;
  1,  1/2, -1/2, -1/6;
The rightmost fraction in row n, that is, the absolute term C(n,0), is the Bernoulli twin number C(n) of A129826(n), i.e., C(n) = A129826(n)/(n+1)!.
If rows are multiplied by (n+1)!, the triangle becomes
    1;
    2,  -1;
    6,   0,  -2;
   24,  12, -12,  -4;
  120, 120, -60, -60, -4;
The sequence a(n) gives the row sums of this triangle. The sums of antidiagonals are 1, 2, 5, 24, 130, 828, 6056.... The first column of the inverse of the triangle is 1, 2, 3, 3, 0, (0 continued).

P. Curtz, Integration numerique ..., Note no. 12 CCSA (later CELAR), 1969. (See A129841, A129696.)
P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A000142, A007318, A048998, A048999, A129724, A129826.

f:= func< n | n le 2 select (-1)^Floor((n+1)/2)/(n+1) else (-1)^n*BernoulliNumber(Floor(n - (1-(-1)^n)/2)) >;
A129378:= func< n | n eq 0 select 1 else Factorial(n+1)*(f(n)+1) >;
[A129378(n): n in [0..30]]; // G. C. Greubel, Feb 01 2024

c[n_?EvenQ] := BernoulliB[n]; c[n_?OddQ] := -BernoulliB[n-1]; c[1] = -1/2; c[2] = -1/3; a[n_] := (n+1)!*(1+c[n]); a[0]=1; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 08 2012, after given formula *)

def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
def A129378(n): return 1 if n==0 else factorial(n+1)*(f(n)+1)
[A129378(n) for n in range(31)] # G. C. Greubel, Feb 01 2024

a(n) = (n+1)!*(1 + C(n)) = A129826(n) + A000142(n+1), n>0.

Edited and extended by R. J. Mathar, Aug 06 2008

A140351 Numerator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

Original entry on oeis.org

1, 0, -1, -1, -1, 1, 1, -1, -3, 3, 5, -5, -691, 691, 35, -35, -3617, 3617, 43867, -43867, -1222277, 1222277, 854513, -854513, -1181820455, 1181820455, 76977927, -76977927, -23749461029, 23749461029, 8615841276005, -8615841276005, -84802531453387, 84802531453387

Offset: 1

Paul Curtz, May 30 2008, Jun 23 2008

The Bernoulli twin number polynomials C(n,x) are defined in A129378.

			The coefficients [x^m]C(n,x) are a table of fractions:
1 ;
-1/2, 1;
-1/3, 0, 1;
-1/6, -1/2, 1/2, 1;
-1/30,-1/2, -1/2, 1, 1;
1/30, -1/6, -1,-1/3, 3/2, 1;
1/42, 1/6, -1/2, -5/3, 0, 2, 1;
-1/42, 1/6, 1/2, -7/6, -5/2, 1/2, 5/2, 1;
-1/30, -1/6, 2/3, 7/6, -7/3, -7/2, 7/6, 3, 1;
1/30, -3/10, -2/3, 2, 7/3, -21/5, -14/3, 2, 7/2, 1;
5/66, 3/10, -3/2, -2, 5, 21/5, -7, -6, 3, 4, 1; ...
This sequence here contains the numerators of the second column.

Cf. A002427, A129378, A129826.

C := proc(n,x) if n = 0 then 1; else add(binomial(n-1,j-1)* bernoulli(j,x),j=1..n) ; expand(%) ; end if ; end proc:
A140351 := proc(n) coeff(C(n,x),x,1) ; numer(%) ; end proc: seq(A140351(n),n=1..80) ; # R. J. Mathar, Nov 22 2009

b[n_, x_] := Coefficient[ Series[ t*E^(x*t)/(E^t - 1), {t, 0, n}], t, n]*n!; c[n_, x_] := Sum[ Binomial[n-1, j-1]*b[j, x], {j, 1, n}]; t[n_, m_] := Coefficient[c[n, x], x, m]; Table[t[n, 1] // Numerator, {n, 1, 34} ] (* Jean-François Alcover, Mar 04 2013 *)
Table[Sum[Binomial[n, k]*(k+1)*BernoulliB[k], {k, 0, n}], {n, 0, 30}] // Numerator (* Vaclav Kotesovec, Oct 05 2016 *)

makelist(num(sum((binomial(n,i)*(i+1)*bern(i)),i,0,n)),n,0,20); /* Vladimir Kruchinin, Oct 05 2016 */

a(n) = numerator(sum(i=0, n, binomial(n,i)*(i+1)*bernfrac(i))); \\ Michel Marcus, Oct 05 2016

a(n) = numerator(Sum_{i=0..n} binomial(n,i)*(i+1)*bernoulli(i)). - Vladimir Kruchinin, Oct 05 2016

Edited and extended by R. J. Mathar, Nov 22 2009

A129724 a(0) = 1; then a(n) = n!(1 - (-1)^nBernoulli(n-1)).

Original entry on oeis.org

1, 2, 3, 7, 24, 116, 720, 5160, 40320, 350784, 3628800, 42940800, 479001600, 4650877440, 87178291200, 2833294464000, 20922789888000, -2166903606067200, 6402373705728000, 6808619561103360000, 2432902008176640000, -26982365129174827008000, 1124000727777607680000

Offset: 0

Paul Curtz, Jun 02 2007

sign

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A027641, A027642, A051716, A051717, A129825, A129826.

Concatenation([1], List([1..25], n-> Factorial(n)*(1 - (-1)^n*Bernoulli(n-1)) )); # G. C. Greubel, Dec 03 2019

[n eq 0 select 1 else Factorial(n)*(1 - (-1)^n*Bernoulli(n-1)): n in [0..25]]; // G. C. Greubel, Dec 03 2019

a:= proc(n)
      if n=0 and n>=0 then 1
    elif n mod 2 = 0 then n!*(1 - bernoulli(n-1))
    else n!*(1 + bernoulli(n-1))
      fi; end;
seq(a(n), n=0..25); # modified by G. C. Greubel, Dec 03 2019

a[0] = 1; a[n_]:= n!*(1-(-1)^n*BernoulliB[n-1]); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Sep 16 2013 *)

a(n) = if(n==0, 1, n!*(1 - (-1)^n*bernfrac(n-1)) ); \\ G. C. Greubel, Dec 03 2019

[1]+[factorial(n)*(1 - (-1)^n*bernoulli(n-1)) for n in (1..25)] # G. C. Greubel, Dec 03 2019

Edited with simpler definition by N. J. A. Sloane, May 25 2008

A140333 Triangle T(n,k) with the coefficient [x^k] (n+1)!* C(n,x), in row n, column k, where C(.,.) are the Bernoulli twin number polynomials of A129378.

Original entry on oeis.org

1, -1, 2, -2, 0, 6, -4, -12, 12, 24, -4, -60, -60, 120, 120, 24, -120, -720, -240, 1080, 720, 120, 840, -2520, -8400, 0, 10080, 5040, -960, 6720, 20160, -47040, -100800, 20160, 100800, 40320, -12096, -60480, 241920, 423360

Offset: 0

Paul Curtz, May 28 2008

The terms at x=0 define the Bernoulli twin numbers, C(n,0)=C(n) = A129826(n)/(n+1)! .
Because the C(n,x) are derived from the Bernoulli polynomials B(n,x) via a binomial transformation and because the odd-indexed Bernoulli numbers are (essentially) zero, the following sum rules for the C(n) emerge (partially in Umbral notation):
For odd C(n): C(2n)=(C-1)^(2n-1), n > 1, C(2n) disappears; example: C(4)=C(4)-3C(3)+3C(2)-C(1).
0r for C(2n+1): (C-1)^2n=0, n >0; example: C(1)-4C(2)+6C(3)-4C(4)+C(5)=0.
With positive coefficients, table
1, 2;
2, 2, 3;
3, 2, 3, 6;
4, 2, 3, 6, 30;
5, 2, 3, 6, 30, -30;
6, 2, 3, 6, 30, -30, -42;
gives C(n). Example: 3C(0)+2C(1)+3C(2)+6C(3)=0. See -A051717(n+1), Bernoulli twin numbers denominators, with from 30 opposite twin.

			1;    C(0,x) = 1
-1, 2;    C(1,x) = -1/2+x
-2, 0, 6;       C(2,x) = -1/3+x^2
-4, -12, 12, 24;      C(3,x) = -1/6 -x/2 +x^2/2 +x^3
-4, -60, -60, 120, 120;

Cf. A129378.

C := proc(n,x) if n =0 then 1; else add( binomial(n-1,j-1)*bernoulli(j,x),j=1..n) ; expand(%) ; end if; end proc:
A140333 := proc(n,k) (n+1)!*C(n,x) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Jun 27 2011

c[0, ] = 1; c[n, x_] := Sum[Binomial[n-1, j-1]*BernoulliB[j, x], {j, 1, n}]; t[n_, k_] := (n+1)!*Coefficient[c[n, x], x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 16 2013 *)

A140812 a(n) = A051717(2n) + A051717(2n+1).

Original entry on oeis.org

3, 9, 60, 84, 60, 132, 5460, 12, 1020, 1596, 660, 276, 5460, 12, 1740, 28644, 1020, 12, 3838380, 12, 27060, 3612, 1380, 564, 92820, 132, 3180, 1596, 1740, 708, 113573460, 12, 1020, 129444, 60, 9372, 280201740, 12, 60, 6636, 460020, 996, 6808620, 12, 122820

Offset: 0

Paul Curtz, Jul 16 2008

nonn

All terms are multiples of 3.

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A129826, A140333.

C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end:
A051717 := proc(n) denom(C(n)) ; end: A140812 := proc(n) A051717(2*n)+A051717(2*n+1) ; end: seq(A140812(n),n=0..80) ; # R. J. Mathar, Jun 28 2009

A051717 := Join[{1}, Denominator[Total /@ Partition[BernoulliB[Range[0, 500]], 2, 1]]]; Join[{3, 9}, Table[2*A051717[[2*n]], {n, 3,30}]] (* G. C. Greubel, Dec 22 2017 *)

Edited and extended by R. J. Mathar, Jun 28 2009

A140334 Triangle read by rows: nonnegative numerators of Bernoulli twin polynomial coefficients on line.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 0, 2, 1, 1, 1, 1, 7, 5, 1, 5, 1, 1, 1, 2, 7, 7, 7, 7, 3, 1, 1, 3, 2, 2, 7, 21, 14, 2, 7, 1, 5, 3, 3, 2, 5, 21, 7, 6, 3, 4, 1

Offset: 0

Paul Curtz, May 28 2008

c[0, x_] = 1; c[n_, x_] := Sum[Binomial[n-1, j-1]*BernoulliB[j, x], {j, 1, n}]; Table[CoefficientList[c[n, x], x], {n, 0, 10}] // Flatten // Abs // Numerator (* Jean-François Alcover, Sep 12 2013 *)

See A129378, A129826 and A140333.
First five increasing polynomials:
1;
-1/2, 1;
-1/3, 0, 1;
-1/6, -1/2, 1/2, 1;
-1/30, -1/2, -1/2, 1, 1;
...

More terms from Jean-François Alcover, Sep 12 2013

cp's OEIS Frontend

A051717 1, followed by denominators of first differences of Bernoulli numbers (B(i)-B(i-1)).

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A051716 Numerators of Bernoulli twin numbers C(n).

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A129825 a(n) = n!*Bernoulli(n-1), n > 2; a(0)=0, a(1)=1, a(2)=1.

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A129378 Row sums of coefficients of Bernoulli twin number polynomials.

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A140351 Numerator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

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A129724 a(0) = 1; then a(n) = n!*(1 - (-1)^n*Bernoulli(n-1)).

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A129724 a(0) = 1; then a(n) = n!(1 - (-1)^nBernoulli(n-1)).