A048998 Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial. Rising powers of x.

1, -1, 2, 1, -6, 6, 0, 12, -36, 24, -4, 0, 120, -240, 120, 0, -120, 0, 1200, -1800, 720, 120, 0, -2520, 0, 12600, -15120, 5040, 0, 6720, 0, -47040, 0, 141120, -141120, 40320, -12096, 0, 241920, 0, -846720, 0, 1693440, -1451520, 362880

Offset

0,3

Comments

See A074909 for generators for the Bernoulli polynomials and connections to the beheaded Pascal triangle and reciprocals of the integers. - Tom Copeland, Nov 17 2014

References

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 9.62.

Links

T. D. Noe, Rows n=0..50 of triangle, flattened

Index entries for sequences related to Bernoulli numbers.

Formula

t*exp(x*t)/(exp(t)-1) = Sum_{n >= 0} B_n(x)*t^n/n!.

a(n,m) = [x^m]((n+1)!*B_n(x)), n>=0, m=0,...,n. - Wolfdieter Lang, Jun 21 2011

Example

B_0(x)=1; B_1(x)=x-1/2; B_2(x)=x^2-x+1/6; B_3(x)=x^3-3*x^2/2+x/2; B_4(x)=x^4-2*x^3+x^2-1/30; ...
Triangle starts:
   1;
  -1,  2;
   1, -6,   6;
   0, 12, -36, 24;
  ...

Maple

A048998 := proc(n, k) coeftayl(bernoulli(n, x), x=0, k) ; (n+1)!*% ; end proc:
seq(seq(A048998(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Jun 27 2011
# second program:
b := proc(n, m, x) option remember; if n = 0 then 1/(m + 1) else
(n + 1) * ((m + 1)*b(n - 1, m + 1, x) - (m + 1 - x)*b(n - 1, m, x)) fi end:
row := n -> seq(coeff(b(n, 0, x), x, k), k = 0..n):
seq(row(n), n = 0..8); # Peter Luschny, Jun 20 2023

Mathematica

Flatten[Table[CoefficientList[(n + 1)! BernoulliB[n, x], x], {n, 0, 10}]] (* T. D. Noe, Jun 21 2011 *)

Crossrefs

Cf. A048999, A074909.

Sequence in context: A376980 A374625 A218853 * A213615 A049019 A133314

Adjacent sequences: A048995 A048996 A048997 * A048999 A049000 A049001

Keyword

sign,easy,nice,tabl

Author

N. J. A. Sloane

Extensions

Added 'Rising powers of x' in name - Wolfdieter Lang, Jun 21 2011

Status

approved