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

0,3

Comments

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.

Links

Table of n, a(n) for n=0..39.

Example

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;

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:
A140333 := proc(n, k) (n+1)!*C(n, x) ; coeftayl(%, x=0, k) ; end proc: # R. J. Mathar, Jun 27 2011

Mathematica

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 *)

Crossrefs

Cf. A129378.

Sequence in context: A338001 A271708 A284983 * A182062 A265326 A245333

Adjacent sequences: A140330 A140331 A140332 * A140334 A140335 A140336

Keyword

sign,less,uned,tabl

Author

Paul Curtz, May 28 2008

Status

approved