A129378 -id:A129378 - cp's OEIS frontend

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.

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 *)

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

Paul Curtz, May 20 2007

sign

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

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

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

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 *)

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

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)).

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

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

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

Paul Curtz, May 30 2008

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

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

Cf. A140351, A129378.

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 *)

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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.

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A129826 Transformed Bernoulli twin numbers.

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

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

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A140352 Denominators of Bernoulli twin numbers polynomial coefficients.

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