A001898 Denominators of Bernoulli polynomials B(n)(x). (Formerly M2014 N0749)

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

0,2

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

Table of n, a(n) for n=0..26.

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.

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).

Example

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.

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 *)

Crossrefs

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

Sequence in context: A266511 A014964 A173181 * A268230 A229628 A002209

Adjacent sequences: A001895 A001896 A001897 * A001899 A001900 A001901

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Entry revised Dec 03 2004

Status

approved