cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A129826 Transformed Bernoulli twin numbers.

Original entry on oeis.org

1, -1, -2, -4, -4, 24, 120, -960, -12096, 120960, 3024000, -36288000, -1576143360, 22066007040, 1525620096000, -24409921536000, -2522591034163200, 45406638614937600, 6686974460694528000, -133739489213890560000, -27033456071346536448000, 594736033569623801856000
Offset: 0

Views

Author

Paul Curtz, May 20 2007

Keywords

Links

Crossrefs

Cf. A027641, A027642, A051717, A051718, A129378, A140351.

Programs

  • Magma
    f:= func< n | n le 2 select (-1)^Floor((n+1)/2)/(n+1) else (-1)^n*BernoulliNumber(Floor(n - (1-(-1)^n)/2)) >;
A129826:= func< n | Factorial(n+1)*f(n) >;
[A129826(n): n in [0..30]]; // G. C. Greubel, Feb 01 2024
  • Mathematica
    c[n_?EvenQ] := BernoulliB[n]; c[n_?OddQ] := -BernoulliB[n-1]; c[1]=-1/2; c[2]=-1/3; a[n_] := (n+1)!*c[n]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 08 2012 *)
  • SageMath
    def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
def A129826(n): return factorial(n+1)*f(n)
[A129826(n) for n in range(31)] # G. C. Greubel, Feb 01 2024

Formula

We define Bernoulli twin numbers C(n) via Bernoulli numbers B(n) = A027641(n)/A027642(n) as C(0)=1, 2C(1)=-1, 3C(2)=-1, C(2n-1)= -B(2n-2) and C(2n)=B(2n), n>1. The sequence is defined as a(n)=(n+1)!*C(n).
a(n) = (n+1)!*C(n), where C(n) = A051718(n)/A051717(n).
E.g.f.: Sum(n>=0) C(n) x^n/n! = 1 + x - x^2/2 + Sum_{n>=1} (B(n) - B(n-1))*x^n/n! = x - x^2/2 + x/(e^x-1) - Integral_{y=0..x} ((y dy)/(e^y-1)).

Extensions

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

A229979 Numerators of interleaved A063524(n) and A002427(n)/A006955(n).

Original entry on oeis.org

0, 1, 1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0
Offset: 0

Views

Author

Paul Curtz, Oct 05 2013

Keywords

Comments

Numerators of Br(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0, -691/210,... complementary Bernoulli numbers.
A164555(n)/A027642(n) is an autosequence of second kind. Its inverse binomial transform is the signed sequence and its main diagonal is the double of the first upper diagonal.
Br(n) is an autosequence of first kind. Its inverse binomial transform is the signed sequence and its main diagonal is A000004=0's.
Br(n) difference table:
0, 1, 1, 1/2, 0, -1/6,...
1, 0, -1/2, -1/2, -1/6, 1/6,... =A140351(n)/A140219(n)
-1, -1/2, 0, 1/3, 1/3, 0,...
1/2, 1/2, 1/3, 0, -1/3, -1/3,...
0, -1/6, -1/3, -1/3, 0, 8/15,...
-1/6, -1/6, 0, 1/3, 8/15, 0,... etc.

Crossrefs

Cf. A050925: a similar sequence, because 2*(n+1)*B(n) and (n+1)*B(n) have the same numerator.

Programs

  • Mathematica
    a[0] = 0; a[1] = a[2] = 1; a[n_] := 2*n*BernoulliB[n-1] // Numerator; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Nov 25 2013 *)

Formula

a(2n)=A063524(n). a(2n+1)=A002427(n).
a(n) = numerators of n * b(n) with b(n)=0 followed by A164555(n)/A027642(n) = 0, 1, 1/2, 1/6, 0,... in A165142(n).
a(n+1) = numerators of Br(n+1) = Br(n) + A140351(n)/A140219(n), a(0)=Br(0)=0.

Extensions

Cross-ref. to A050925 by Jean-François Alcover, Dec 09 2013

A140219 Denominator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 6, 6, 10, 10, 6, 6, 210, 210, 2, 2, 30, 30, 42, 42, 110, 110, 6, 6, 546, 546, 2, 2, 30, 30, 462, 462, 170, 170, 6, 6, 51870, 51870, 2, 2, 330, 330, 42, 42, 46, 46, 6, 6, 6630, 6630, 22, 22, 30, 30, 798, 798, 290
Offset: 0

Views

Author

Paul Curtz, Jun 23 2008

Keywords

Comments

See A140351 for the main part of the documentation.

Crossrefs

Cf. A002427, A006955, A048594, A140351 (numerators).

Programs

  • Maple
    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:
A140219 := proc(n) coeff(C(n, x), x, 1) ; denom(%) ; end proc:
seq(A140219(n), n=1..80) ; # R. J. Mathar, Sep 22 2011
  • Mathematica
    Table[Sum[Binomial[n, k]*(k+1)*BernoulliB[k], {k, 0, n}], {n, 0, 60}] // Denominator (* Vaclav Kotesovec, Oct 05 2016 *)
  • Maxima
    makelist(denom(sum((binomial(n, i)*(i+1)*bern(i)), i, 0, n)), n, 0, 20); /* Vladimir Kruchinin, Oct 05 2016 */
  • PARI
    a(n) = denominator(sum(i=0, n, binomial(n,i)*(i+1)*bernfrac(i))); \\ Michel Marcus, Oct 05 2016

Formula

a(n) = denominator(Sum_{i=0..n} binomial(n,i)*(i+1)*bern(i)). - Vladimir Kruchinin, Oct 05 2016
a(n) = A006955(floor(n/2)). - Georg Fischer, Nov 29 2022

A140352 Denominators of Bernoulli twin numbers polynomial coefficients.

Original entry on oeis.org

1, -2, 1, -3, 1, -6, -2, 2, 1, -30, -2, -2, 1, 1, 30, -6, -1, -3, 2, 1, 42, 6, -2, -3, 1, 1, -42, 6, 2, -6, -2, 2, 2, 1, -30, -6, 3, 6, -3, -2, 6, 1, 1, 30, -10, -3, 1, 3, -5, -3, 1, 2, 1, 66, 10, -2, -1, 1, 5, -1, -1, 1, 1, 1
Offset: 0

Views

Author

Paul Curtz, May 30 2008

Keywords

Comments

See numerators A140334, when numerator is 0, no entry is entered here.

Examples

			Triangle starts:
1;
-2, 1;
-3, 1;
-6, -2, 2, 1;

Crossrefs

Cf. A140351, A129378.

Programs

  • Mathematica
    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 // Select[#, # != 0 &]& // Sign[#]*Denominator[#]& (* Jean-François Alcover, Sep 12 2013 *)

A242246 Numerators of n*A164555(n-1)/A027642(n-1).

Original entry on oeis.org

0, 1, 1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615641276005, 0, -84802531453387, 0, 90219075042845, 0
Offset: 0

Views

Author

Paul Curtz, May 09 2014

Keywords

Comments

First multiplied shifted (second) Bernoulli numbers.
A164555(n-1)/A027642(n-1) = 0 followed by (A164555(n)/A027642(n)=1, 1/2, 1/6,...) = f(n) = 0, 1, 1/2, 1/6, 0,... .
f(n+1) - f(n) = A051716(n)/A051717(n).
Generally we consider a transform applied to the autosequences of first or second kind. An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. It is of the first kind if the main diagonal is A000004=0's. It is of the second kind if the main diagonal is the first upper diagonal multiplied by 2. A000045(n) is an autosequence of the first kind. A164555(n)/A027642(n) is an autosequence of the second kind. See A190339 (and A241269).
Here we apply the transform to the Bernoulli numbers A164555(n)/A027642(n).
We take n*(0 followed by A164555(n)/A027642(n)).
Hence the autosequence of first kind
TB1(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, O, -691/210,.. .
a(n) are the numerators.
The first seven rows of the differencece table of TB1(n) are
0, 1, 1, 1/2, 0, - 1/6, 0, 1/6,...
1, 0, -1/2, -1/2, -1/6, 1/6, 1/6, -1/6,... =A140351(n+1)/b(n+1)
-1, -1/2, 0, 1/3, 1/3, 0, -1/3, -2/15,...
1/2, 1/2, 1/3, 0, -1/3, -1/3, 1/5, 11/15,...
0, -1/6, -1/3, -1/3, 0, 8/15, 8/15, -4/5,...
-1/6, -1/6, 0, 1/3, 8/15, 0, -4/3, -4/3,...
0, 1/6, 1/3, 1/5, -8/15, -4/3, 0, 512/105,... .
First and second upper diagonals: 1, -1/2, 1/3, -1/3, 8/15, -4/3, 512/105,... .
Sum of the antidiagonals:
0, 1, 1, 0, -1/2, 0, 1/2, 0, -5/6, 0, 13/6, 0, -49/6, 0,... .
(Note that the same transform applied to the second fractional Euler numbers A198631(n)/A006519(n+1) yields the Genocchi numbers -A226158(n)).
This transform can be continued:
TB2(n) = n*(0 followed by TB1(n)) =
0, 0, 2, 3, 2, 0, -1, 0, 4/3, 0, -3, 0, 10, 0, -691/15, 0, 280, 0,...
is an autosequence of second kind.
TB3(n) = 0, 0, 0, 6, 12, 10, 0, -7, 0, 12, 0, -33, 0, 130, 0, 691, 0,...
is apparently an integer autosequence of the first kind.

Crossrefs

Cf. A199969 (autosequence).

Formula

a(n) = 0 followed by (A050925(n) = 1, -1, 1, 0,... ) with 1 instead of -1.
a(2n) = A063524(n). a(2n+1) = A002427(n).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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