A087617 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)).

1, 2, 82, 10572, 2860662, 1330910844, 947622146676, 957663025230936, 1303349182536886566, 2298001401440208011756, 5095053865489946980238428, 13874003700656227505945514920, 45517269584820569745186971856060

Offset: 0

Author

Juan Arias-de-Reyna, Sep 12 2003

Keywords

• easy
• nonn
• nice

Comments

Gabcke conjectured and Juan Arias-de-Reyna proved that the terms are integers.

They also appear as the coefficients of the asymptotic expansion Sum_{n>=0} a(n) tau^(4n) of the function Re log Gamma(1/4 + i*t/2) + Pi*t/4 + (1/4)*log(t/2) - log(sqrt(2*Pi)), where tau = (1/2)*sqrt(2t).

References

• W. Gabcke, Neue Herleitung und explizite Restabschaetzung der Riemann-Siegel-Formel. Dissertation, Univ. Goettingen (1979).

Links

• J. Arias de Reyna, Dynamical zeta functions and Kummer congruences.

Programs

Mathematica

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