Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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