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%I M2014 N0749 #25 Jul 25 2022 04:04:49
%S 1,2,12,8,240,96,4032,1152,34560,7680,101376,18432,50319360,7741440,
%T 6635520,884736,451215360,53084160,42361159680,4459069440,
%U 1471492915200,140142182400,1758147379200,152882380800,417368899584000,33389511966720,15410543984640
%N Denominators of Bernoulli polynomials B(n)(x).
%D F. N. David, Probability Theory for Statistical Methods, Cambridge, 1949; see pp. 103-104. [There is an error in the recurrence for B_s^{(r)}.]
%D N. E. Nørlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 459.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F These Bernoulli polynomials B(s) = B(s)(x) are defined by: B(0) = 1; B(s) = (-x/s)*Sum_{t=1..s} (-1)^t*binomial(s, t)*Bernoulli(t)*B(s-t), where Bernoulli(t) are the usual Bernoulli numbers A027641/A027642. Also B(s)(1) = Bernoulli(s).
%e The Bernoulli polynomials B(0)(x) through B(6)(x) are:
%e 1;
%e -(1/2)*x;
%e (1/12)*(3*x-1)*x;
%e -(1/8)*(x-1)*x^2;
%e (1/240)*(15*x^3-30*x^2+5*x+2)*x;
%e -(1/96)*(x-1)*(3*x^2-7*x-2)*x^2;
%e (1/4032)*(63*x^5-315*x^4+315*x^3+91*x^2-42*x-16)*x.
%p B:=bernoulli; b:=proc(s) option remember; local t; global r; if s=0 then RETURN(1); fi; expand((-r/s)*add( (-1)^t*binomial(s,t)*B(t)*b(s-t),t=1..s)); end; [seq(denom(b(n)),n=0..30)];
%t B[s_] := B[s] = If[s == 0, 1, (-x/s)*Sum[(-1)^t*Binomial[s, t]*
%t BernoulliB[t]*B[s - t], {t, 1, s}]] // Factor;
%t a[n_] := If[n == 0, 1, B[n] // First // Denominator];
%t Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Jul 24 2022 *)
%Y Cf. A027641, A027642, A100615, A100616, A100655.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Entry revised Dec 03 2004
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