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Search: id:A087617
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| A087617 |
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Gabcke sequence: a(0)=1; (n+1) a(n+1) = Sum_{k=0..n} 2^(4k+1) |E(2k+2)| a(n-k), where |E(2k+2)| are Euler numbers (E(2k)=(-1)^k A000364(k)). |
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+0
1
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| 1, 2, 82, 10572, 2860662, 1330910844, 947622146676, 957663025230936, 1303349182536886566, 2298001401440208011756, 5095053865489946980238428, 13874003700656227505945514920, 45517269584820569745186971856060
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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The recurrence does not imply that these numbers are integers. Gabcke conjectured that they are integers. This is proved in Arias de Reyna, 'Dynamical zeta functions and Kummer congruences'. They also appear as the coefficients of the asymptotic expansion Sum a(n) tau^(4n) n=0...infinity of the function Re log Gamma(1/4 +it/2) + Pi t/4 +(1/4)log(t/2) -log sqrt(2Pi), where tau=1/2sqrt(2t)
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REFERENCES
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W. Gabcke, Neue Herleitung und explizite Restabschaetzung der Riemann-Siegel-Formel. Dissertation, Univ. Goettingen (1979)
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LINKS
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J. Arias de Reyna, Dynamical zeta functions and Kummer congruences .
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MATHEMATICA
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lambda[0] = 1; lambda[n_] := lambda[n] = Sum[2^(4 k + 1) Abs[EulerE[2k + 2]]lambda[n - 1 - k], {k, 0, n - 1}]/n
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CROSSREFS
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Sequence in context: A061994 A093666 A063270 this_sequence A140157 A139867 A090434
Adjacent sequences: A087614 A087615 A087616 this_sequence A087618 A087619 A087620
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KEYWORD
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easy,nonn,nice
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AUTHOR
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Juan Arias de Reyna (arias(AT)us.es), Sep 12 2003
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