OFFSET
0,2
COMMENTS
See Nørlund for precise definition.
The 'higher order Bernoulli numbers' considered here are the values of the 'higher order Bernoulli polynomials' evaluated at x=1 (and not at x=0, which would make things boring as x is a factor of these polynomials for n>0). This can be seen as an argument that the definition of the classical Bernoulli numbers as the values of the classical Bernoulli polynomials at x=1 better fits into the general picture than the often used definition as the values at x=0. - Peter Luschny, Oct 01 2016
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
LINKS
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer, 1924, p. 459.
EXAMPLE
From Peter Luschny, Oct 01 2016: (Start)
The sequence of polynomials starts:
1,
(1/12*(3*x-1))*x,
(1/240*(15*x^3-30*x^2+5*x+2))*x,
(1/4032*(63*x^5-315*x^4+315*x^3+91*x^2-42*x-16))*x,
(1/34560*(135*x^7-1260*x^6+3150*x^5-840*x^4-2345*x^3-540*x^2+404*x+144))*x. (End)
MAPLE
B := proc(v, n) option remember; `if`(v = 0, 1,
simplify(-(n/v)*add((-1)^s*binomial(v, s)*bernoulli(s)*B(v-s, n), s=1..v))) end:
A213449 := n -> denom(B(2*n, k)):
seq(A213449(n), n=0..19); # Peter Luschny, Oct 01 2016
MATHEMATICA
Table[NorlundB[2n, x] // Together // Denominator, {n, 0, 19}] (* Jean-François Alcover, Jun 29 2019 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Jun 12 2012
EXTENSIONS
Name corrected and more terms added by Peter Luschny, Oct 01 2016
STATUS
approved