A213449 Denominators of higher order Bernoulli numbers. (Formerly N2089)

1, 12, 240, 4032, 34560, 101376, 50319360, 6635520, 451215360, 42361159680, 1471492915200, 1758147379200, 417368899584000, 15410543984640, 141874849382400, 28026642660065280, 922166952040857600, 19725496300339200, 2163255728265599385600, 36926129074234982400

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

Table of n, a(n) for n=0..19.

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

Cf. A000367 (numerators of the polynomials evaluated at x=1 at even indices).

Bisection (even indices) of A001898.

Sequence in context: A033469 A012544 A009052 * A012303 A119837 A012538

Adjacent sequences: A213446 A213447 A213448 * A213450 A213451 A213452

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