A000411 Generalized tangent numbers d(6,n). (Formerly M4312 N1805)

6, 522, 152166, 93241002, 97949265606, 157201459863882, 357802951084619046, 1096291279711115037162, 4350684698032741048452486, 21709332137467778453687752842, 133032729004732721625426681085926, 982136301747914281420205946546842922, 8597768767880274820173388403096814519366

Offset

1,1

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

Lars Blomberg, Table of n, a(n) for n = 1..184

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.

D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699.

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Formula

a(n) = (2*n-1)! * [x^(2*n-1)] 2*sin(3*x) / (2*cos(4*x) - 1). - F. Chapoton, Oct 06 2020

a(n) = (2*n-1)!*[x^(2*n-1)](sec(6*x)*(sin(x) + sin(5*x))). - Peter Luschny, Nov 21 2021

Maple

egf := sec(6*x)*(sin(x) + sin(5*x)): ser := series(egf, x, 24):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..12); # Peter Luschny, Nov 21 2021

Mathematica

nmax = 15; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[-a, 2 k + 1]/(2 k + 1)^s, {k, 0, km}]; d[a_ /; a > 1, n_, km_] := (2 n - 1)! L[-a, 2 n, km] (2 a/Pi)^(2 n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[6, n, km], {n, 1, nmax}]; dd[km0]; dd[km = 2 km0]; While[dd[km] != dd[km/2, km = 2 km]]; A000411 = dd[km] (* Jean-François Alcover, Feb 08 2016 *)

Prog

(Sage)
t = PowerSeriesRing(QQ, 't', default_prec=24).gen()
f = 2 * sin(3 * t) / (2 * cos(4 * t) - 1)
f.egf_to_ogf().list()[1::2] # F. Chapoton, Oct 06 2020

Crossrefs

Cf. A000192, A000320, A001587, A349264.

Sequence in context: A250391 A003395 A222607 * A180431 A202967 A230330

Adjacent sequences: A000408 A000409 A000410 * A000412 A000413 A000414

Keyword

nonn

Author

N. J. A. Sloane

Extensions

a(10)-a(12) from Lars Blomberg, Sep 07 2015

Status

approved