%I M1715 N0679 #60 Nov 22 2021 04:03:38

%S 2,6,46,522,7970,152166,3487246,93241002,2849229890,97949265606,

%T 3741386059246,157201459863882,7205584123783010,357802951084619046,

%U 19133892392367261646,1096291279711115037162,67000387673723462963330,4350684698032741048452486,299131045427247559446422446

%N Generalized Euler numbers.

%C These numbers are related to the values at negative integers of the L-functions for two primitive Dirichlet characters of conductor 24. - _F. Chapoton_, Oct 05 2020

%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 Lars Blomberg, <a href="/A001587/b001587.txt">Table of n, a(n) for n = 0..199</a>

%H LMFDB, <a href="https://www.lmfdb.org/Character/Dirichlet/24/5">character 24.5</a>

%H LMFDB, <a href="https://www.lmfdb.org/Character/Dirichlet/24/11">character 24.11</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/S0025-5718-68-99652-X">Corrigendum: 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]

%F E.g.f.: 2 (sin(3 x) + cos(3 x)) / (2 cos(4 x) - 1). - _F. Chapoton_, Oct 06 2020

%F a(n) ~ 2^(2*n + 2) * 3^(n + 1/2) * n^(n + 1/2) / (exp(n) * Pi^(n + 1/2)). - _Vaclav Kotesovec_, Nov 05 2021

%F a(n) = n!*[x^n](sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x))). - _Peter Luschny_, Nov 21 2021

%p egf := sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)): ser := series(egf, x, 20): seq(n!*coeff(ser, x, n), n = 0..17); # _Peter Luschny_, Nov 21 2021

%o (Sage)

%o t = PowerSeriesRing(QQ, 't').gen()

%o f = 2 * (sin(3 * t) + cos(3 * t)) / (2 * cos(4 * t) - 1)

%o f.egf_to_ogf().list() # _F. Chapoton_, Oct 06 2020

%Y Bisections are A000192 and A000411. Overview in A349264.

%Y Similar sequences: A000111, A225147.

%K nonn

%O 0,1

%A _N. J. A. Sloane_

%E a(11)-a(14) from _Lars Blomberg_, Sep 10 2015