A352977 Expansion of e.g.f. cos(2x) cos(3x) / cos(6x) (even powers only).

1, 23, 3985, 1743623, 1424614945, 1870693029623, 3602792061891505, 9566946196183630823, 33500193836861731481665, 149565522713623779723211223, 829235405016410370201483113425, 5589623533324449496004527793434823, 45017811997394066193946619670380594785

Offset

0,2

Comments

Only terms of even index are given. Terms of odd index are zero.

Links

Table of n, a(n) for n=0..12.

D. Choi, S. Lim and R. C. Rhoades, Mock modular forms and quantum modular forms, Proc. Amer. Math. Soc. 144 (2016), 2337-2349. (See page 2341.)

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices. IV. The Mass Formula, Proc. Roy. Soc. London Ser. A 419 (1988), no. 1857, 259-286. (See table 6.)

M. Monks, Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts, Proc. Amer. Math. Soc. 138 (2010), no. 2, 481-494. (See page 485.)

D. Shanks and J. W. Wrench, The calculation of certain Dirichlet series, Math. Comp. 17 (1963), 136-154. (See line 6 of Table 1.)

Formula

E.g.f.: cos(2*x) * cos(3*x) / cos(6*x).

From Peter Luschny, Apr 13 2022: (Start)

E.g.f.: (cos(x) + cos(5*x))*sec(6*x) / 2, even powers only.

a(n) = A000192(n)/2. (End)

a(n) ~ 2^(6*n + 3/2) * 3^(2*n + 1/2) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 15 2022

Maple

egf := (cos(x) + cos(5*x))*sec(6*x) / 2: ser := series(egf, x, 32):
seq(n!*coeff(ser, x , n), n = 0..24, 2); # Peter Luschny, Apr 13 2022

Prog

(Sage)
x = PowerSeriesRing(QQ, 'x', default_prec=30).gen()
f = cos(2*x) * cos(3*x) / cos(6*x)
[cf for cf in f.egf_to_ogf() if cf]
(PARI) my(x='x+O('x^30)); select(x->(x>0), Vec(serlaplace(cos(2*x)*cos(3*x)/cos(6*x)))) \\ Michel Marcus, Apr 13 2022

Crossrefs

Intermediate case between A002437 and A349429.

Cf. A000192.

Sequence in context: A233143 A134798 A368137 * A308458 A103443 A059000

Adjacent sequences: A352974 A352975 A352976 * A352978 A352979 A352980

Keyword

nonn,easy

Author

F. Chapoton, Apr 13 2022

Status

approved