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%I M4312 N1805 #46 Nov 22 2021 04:03:30
%S 6,522,152166,93241002,97949265606,157201459863882,357802951084619046,
%T 1096291279711115037162,4350684698032741048452486,
%U 21709332137467778453687752842,133032729004732721625426681085926,982136301747914281420205946546842922,8597768767880274820173388403096814519366
%N Generalized tangent numbers d(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 Lars Blomberg, <a href="/A000411/b000411.txt">Table of n, a(n) for n = 1..184</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]
%F a(n) = (2*n-1)! * [x^(2*n-1)] 2*sin(3*x) / (2*cos(4*x) - 1). - _F. Chapoton_, Oct 06 2020
%F a(n) = (2*n-1)!*[x^(2*n-1)](sec(6*x)*(sin(x) + sin(5*x))). - _Peter Luschny_, Nov 21 2021
%p egf := sec(6*x)*(sin(x) + sin(5*x)): ser := series(egf, x, 24):
%p seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..12); # _Peter Luschny_, Nov 21 2021
%t nmax = 15; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[-a, 2 k + 1]/(2 k + 1)^s, {k, 0, km}]; d[a_ /; a > 1, n_, km_] := (2 n - 1)! L[-a, 2 n, km] (2 a/Pi)^(2 n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[6, n, km], {n, 1, nmax}]; dd[km0]; dd[km = 2 km0]; While[dd[km] != dd[km/2, km = 2 km]]; A000411 = dd[km] (* _Jean-François Alcover_, Feb 08 2016 *)
%o (Sage)
%o t = PowerSeriesRing(QQ, 't', default_prec=24).gen()
%o f = 2 * sin(3 * t) / (2 * cos(4 * t) - 1)
%o f.egf_to_ogf().list()[1::2] # _F. Chapoton_, Oct 06 2020
%Y Cf. A000192, A000320, A001587, A349264.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E a(10)-a(12) from _Lars Blomberg_, Sep 07 2015
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