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%I M2166 N0864 #72 Nov 22 2021 04:03:03
%S 2,46,3362,515086,135274562,54276473326,30884386347362,
%T 23657073914466766,23471059057478981762,29279357851856595135406,
%U 44855282210826271011257762,82787899853638102222862479246,181184428895772987376073015175362,463938847087789978515380344866258286
%N Generalized tangent numbers d(3, 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 Peter Luschny, <a href="/A000191/b000191.txt">Table of n, a(n) for n = 0..250</a>
%H Daniel 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 Daniel 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 Daniel Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 690. [Annotated scanned copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TangentNumber.html">Tangent Number</a>.
%F a(n) = 2*A002439(n). - _N. J. A. Sloane_, Nov 06 2009
%F E.g.f.: (2*sin(t))/(2*cos(2*t) - 1), odd terms only. - _Peter Luschny_, Oct 17 2020
%F Alternative form for e.g.f.: a(n) = (2*n+1)!*[x^(2*n)](sqrt(3)/(6*x))*(sec(x + Pi/3) + sec(x + 2*Pi/3)). - _Peter Bala_, Nov 16 2020
%F a(n) = (-1)^(n+1)*6^(2*n+1)*euler(2*n+1, 1/6). - _Peter Luschny_, Nov 26 2020
%p gf := (2*sin(t))/(2*cos(2*t) - 1): ser := series(gf, t, 26):
%p seq((2*n+1)!*coeff(ser, t, 2*n+1), n=0..23); # _Peter Luschny_, Oct 17 2020
%p a := n -> (-1)^n*(-6)^(2*n+1)*euler(2*n+1, 1/6):
%p seq(a(n), n = 0..13); # _Peter Luschny_, Nov 26 2020
%t (* Formulas from D. Shanks, see link, p. 690. *)
%t L[ a_, s_, t_:10000 ] := Plus@@Table[ N[ JacobiSymbol[ -a, 2k+1 ](2k+1)^(-s), 30 ], {k, 0, t} ]; d[ a_, n_, t_:10000 ] := (2n-1)!/Sqrt[ a ](2a/Pi)^(2n)L[ -a, 2n, t ] (* _Eric W. Weisstein_, Aug 30 2001 *)
%Y Cf. A000187, A000192, A002437, A002439, A156172, A235606 (row 3).
%Y Cf. A000436, A007289, overview in A349264.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E More terms from _Eric W. Weisstein_, Aug 30 2001
%E Offset set to 0 by _Peter Luschny_, Nov 26 2020
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