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 A000191 Generalized tangent numbers d(3, n). (Formerly M2166 N0864) 12
 2, 46, 3362, 515086, 135274562, 54276473326, 30884386347362, 23657073914466766, 23471059057478981762, 29279357851856595135406, 44855282210826271011257762, 82787899853638102222862479246, 181184428895772987376073015175362, 463938847087789978515380344866258286 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Peter Luschny, Table of n, a(n) for n = 0..250 Daniel Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694. Daniel Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699. Daniel Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 690. [Annotated scanned copy] Eric Weisstein's World of Mathematics, Tangent Number. FORMULA a(n) = 2*A002439(n). - N. J. A. Sloane, Nov 06 2009 E.g.f.: (2*sin(t))/(2*cos(2*t) - 1), odd terms only. - Peter Luschny, Oct 17 2020 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 a(n) = (-1)^(n+1)*6^(2*n+1)*euler(2*n+1, 1/6). - Peter Luschny, Nov 26 2020 MAPLE gf := (2*sin(t))/(2*cos(2*t) - 1): ser := series(gf, t, 26): seq((2*n+1)!*coeff(ser, t, 2*n+1), n=0..23); # Peter Luschny, Oct 17 2020 a := n -> (-1)^n*(-6)^(2*n+1)*euler(2*n+1, 1/6): seq(a(n), n = 0..13); # Peter Luschny, Nov 26 2020 MATHEMATICA (* Formulas from D. Shanks, see link, p. 690. *) 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 *) CROSSREFS Cf. A000187, A000192, A002437, A002439, A156172, A235606 (row 3). Cf. A000436, A007289, overview in A349264. Sequence in context: A173586 A074041 A277554 * A000192 A196197 A273380 Adjacent sequences: A000188 A000189 A000190 * A000192 A000193 A000194 KEYWORD nonn,easy AUTHOR N. J. A. Sloane EXTENSIONS More terms from Eric W. Weisstein, Aug 30 2001 Offset set to 0 by Peter Luschny, Nov 26 2020 STATUS approved

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Last modified September 22 07:52 EDT 2023. Contains 365519 sequences. (Running on oeis4.)