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A213449 Denominators of higher order Bernoulli numbers.

%I A213449 N2089 #39 Jun 29 2019 06:30:52
%S A213449 1,12,240,4032,34560,101376,50319360,6635520,451215360,42361159680,
%T A213449 1471492915200,1758147379200,417368899584000,15410543984640,
%U A213449 141874849382400,28026642660065280,922166952040857600,19725496300339200,2163255728265599385600,36926129074234982400
%N A213449 Denominators of higher order Bernoulli numbers.
%C A213449 See Nørlund for precise definition.
%C A213449 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
%D A213449 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H A213449 N. E. Nørlund, <a href="https://eudml.org/doc/204170">Vorlesungen über Differenzenrechnung</a>, Springer, 1924, p. 459.
%e A213449 From _Peter Luschny_, Oct 01 2016: (Start)
%e A213449 The sequence of polynomials starts:
%e A213449 1,
%e A213449 (1/12*(3*x-1))*x,
%e A213449 (1/240*(15*x^3-30*x^2+5*x+2))*x,
%e A213449 (1/4032*(63*x^5-315*x^4+315*x^3+91*x^2-42*x-16))*x,
%e A213449 (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)
%p A213449 B := proc(v,n) option remember; `if`(v = 0,1,
%p A213449 simplify(-(n/v)*add((-1)^s*binomial(v,s)*bernoulli(s)*B(v-s,n),s=1..v))) end:
%p A213449 A213449 := n -> denom(B(2*n, k)):
%p A213449 seq(A213449(n), n=0..19); # _Peter Luschny_, Oct 01 2016
%t A213449 Table[NorlundB[2n, x] // Together // Denominator, {n, 0, 19}] (* _Jean-François Alcover_, Jun 29 2019 *)
%Y A213449 Cf. A000367 (numerators of the polynomials evaluated at x=1 at even indices).
%Y A213449 Bisection (even indices) of A001898.
%K A213449 nonn,frac
%O A213449 0,2
%A A213449 _N. J. A. Sloane_, Jun 12 2012
%E A213449 Name corrected and more terms added by _Peter Luschny_, Oct 01 2016