A000191 Generalized tangent numbers d(3, n).
2, 46, 3362, 515086, 135274562, 54276473326, 30884386347362, 23657073914466766, 23471059057478981762, 29279357851856595135406, 44855282210826271011257762, 82787899853638102222862479246, 181184428895772987376073015175362, 463938847087789978515380344866258286
Offset: 0
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.
Crossrefs
Programs
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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
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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 *)
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
Extensions
More terms from Eric W. Weisstein, Aug 30 2001
Offset set to 0 by Peter Luschny, Nov 26 2020