A000191 Generalized tangent numbers d(3, n). (Formerly M2166 N0864)

2, 46, 3362, 515086, 135274562, 54276473326, 30884386347362, 23657073914466766, 23471059057478981762, 29279357851856595135406, 44855282210826271011257762, 82787899853638102222862479246, 181184428895772987376073015175362, 463938847087789978515380344866258286

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