A001898 Denominators of Bernoulli polynomials B(n)(x).

1, 2, 12, 8, 240, 96, 4032, 1152, 34560, 7680, 101376, 18432, 50319360, 7741440, 6635520, 884736, 451215360, 53084160, 42361159680, 4459069440, 1471492915200, 140142182400, 1758147379200, 152882380800, 417368899584000, 33389511966720, 15410543984640

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			The Bernoulli polynomials B(0)(x) through B(6)(x) are:
1;
-(1/2)*x;
(1/12)*(3*x-1)*x;
-(1/8)*(x-1)*x^2;
(1/240)*(15*x^3-30*x^2+5*x+2)*x;
-(1/96)*(x-1)*(3*x^2-7*x-2)*x^2;
(1/4032)*(63*x^5-315*x^4+315*x^3+91*x^2-42*x-16)*x.
		

References

• 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)}.]
• N. E. Nørlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 459.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A027641, A027642, A100615, A100616, A100655.

Programs

Maple

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)];

Mathematica

B[s_] := B[s] = If[s == 0, 1, (-x/s)*Sum[(-1)^t*Binomial[s, t]*
   BernoulliB[t]*B[s - t], {t, 1, s}]] // Factor;
a[n_] := If[n == 0, 1, B[n] // First // Denominator];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 24 2022 *)

Formula

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).

Extensions

Entry revised Dec 03 2004