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
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
KEYWORD
AUTHOR
EXTENSIONS
Added 'Rising powers of x' in name - Wolfdieter Lang, Jun 21 2011
STATUS
approved