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Search: id:A001898
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| A001898 |
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Denominators of Bernoulli polynomials B(n)(x).
(Formerly M2014 N0749)
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+0
5
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| 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
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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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. N\"{o}rlund, Vorlesungen \"{u}ber Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 459.
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LINKS
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Index entries for sequences related to Bernoulli numbers.
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FORMULA
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These Bernoulli polynomials B(s) = B(s)(x) are defined by: B(0) = 1; B(s) = (-x/s)*Sum( (-1)^t*binomial(s, t)*Bernoulli(t)*B(s-t), t=1..s)), where Bernoulli(t) are the usual Bernoulli numbers A027641/A027642. Also B(s)(1) = Bernoulli(s).
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EXAMPLE
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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
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MAPLE
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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)];
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CROSSREFS
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Cf. A027641, A027642, A100615, A100616, A100655.
Sequence in context: A107414 A133437 A014964 this_sequence A002209 A100654 A081468
Adjacent sequences: A001895 A001896 A001897 this_sequence A001899 A001900 A001901
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KEYWORD
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nonn
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AUTHOR
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njas
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EXTENSIONS
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Entry revised Dec 03, 2004.
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