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A006569
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Numerators of generalized Bernoulli numbers.
(Formerly M3731)
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2
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1, -1, 1, 1, -1, -5, -1, 7, 13, -307, -479, 1837, 100921, 15587, -23737, -5729723, 14731223, 9129833, 2722952839, -4700745901, -1556262845, 190717213397, 24684889339847, -50242799489, -148437433077277, -8592042383621, 221844330989749, 176585172615885307, -9245931549625447
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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REFERENCES
| F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J., 34 (1967), 599-615.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, Page 7, 2nd table is identical to A006569/A006568.
Abdul Hassen and Hieu D. Nguyen, Hypergeometric Zeta Functions, Sep 27 2005.
Index entries for sequences related to Bernoulli numbers.
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FORMULA
| Recurrence relation: bernoulli(n+1)=a[n+1]-sum(binomial(n+1, r)*bernoulli(r)*a[n+2-r], r=1..n+1); a[0]=1. (p. 603 of the Howard reference). - Deutsch - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 23 2005
E.g.f. for fractions: x^2/2 / (e^x-1-x). [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 04 2009]
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MAPLE
| eq:=n->bernoulli(n+1)=a[n+1]-sum(binomial(n+1, r)*bernoulli(r)*a[n+2-r], r=1..n+1): a[0]:=1:for n from 0 to 28 do a[n+1]:=solve(eq(n), a[n+1]) od: seq(numer(a[n]), n=0..29); (Deutsch)
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CROSSREFS
| Cf. A006568.
Cf. A132092-A132106.
Sequence in context: A100122 A001945 A051854 * A193860 A126155 A021197
Adjacent sequences: A006566 A006567 A006568 * A006570 A006571 A006572
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KEYWORD
| frac,sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 23 2005
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