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%I M2167 N0865 #51 Apr 15 2022 04:56:33
%S 2,46,7970,3487246,2849229890,3741386059246,7205584123783010,
%T 19133892392367261646,67000387673723462963330,
%U 299131045427247559446422446,1658470810032820740402966226850,11179247066648898992009055586869646,90035623994788132387893239340761189570
%N Generalized Euler numbers c(6,n).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Sean A. Irvine, <a href="/A000192/b000192.txt">Table of n, a(n) for n = 0..250</a>
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967) 689-694.
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1968-0227093-9">Corrigenda to: "Generalized Euler and class numbers"</a>, Math. Comp. 22 (1968), 699.
%H D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerNumber.html">Euler Number</a>.
%F E.g.f.: 2*cos(3*x) / (2*cos(4*x) - 1). - _F. Chapoton_, Oct 06 2020
%F a(n) = (2*n)!*[x^(2*n)](sec(6*x)*(cos(x) + cos(5*x))). - _Peter Luschny_, Nov 21 2021
%F a(n) ~ 2^(6*n + 5/2) * 3^(2*n + 1/2) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - _Vaclav Kotesovec_, Apr 15 2022
%p egf := sec(6*x)*(cos(x) + cos(5*x)): ser := series(egf, x, 24):
%p seq((2*n)!*coeff(ser, x, 2*n), n = 0..10); # _Peter Luschny_, Nov 21 2021
%t L[ a_, s_, t_:10000 ] := Plus@@Table[ N[ JacobiSymbol[ -a, 2k+1 ](2k+1)^(-s), 30 ], {k, 0, t} ]; c[ a_, n_, t_:10000 ] := (2n)!/Sqrt[ a ](2a/Pi)^(2n+1)L[ a, 2n+1, t ] (* _Eric W. Weisstein_, Aug 30 2001 *)
%o (Sage)
%o t = PowerSeriesRing(QQ, 't', default_prec=24).gen()
%o f = 2 * cos(3 * t) / (2 * cos(4 * t) - 1)
%o f.egf_to_ogf().list()[::2] # _F. Chapoton_, Oct 06 2020
%Y Cf. A000187, A000191, A000411, A001587, A349264.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E More terms from _Eric W. Weisstein_, Aug 30 2001
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