A000411 Generalized tangent numbers d(6,n).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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]

Crossrefs

Cf. A000192, A000320, A001587, A349264.

Programs

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 *)

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

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

Extensions

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