A213449 Denominators of higher order Bernoulli numbers.

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

Offset: 0

Author

N. J. A. Sloane, Jun 12 2012

Keywords

• nonn
• frac

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

Examples

			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)
		

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.

Crossrefs

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

Bisection (even indices) of A001898.

Programs

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 *)

Extensions

Name corrected and more terms added by Peter Luschny, Oct 01 2016