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%I #23 Apr 15 2022 14:57:14
%S 1,23,3985,1743623,1424614945,1870693029623,3602792061891505,
%T 9566946196183630823,33500193836861731481665,
%U 149565522713623779723211223,829235405016410370201483113425,5589623533324449496004527793434823,45017811997394066193946619670380594785
%N Expansion of e.g.f. cos(2x) cos(3x) / cos(6x) (even powers only).
%C Only terms of even index are given. Terms of odd index are zero.
%H D. Choi, S. Lim and R. C. Rhoades, <a href="https://doi.org/10.1090/proc/12907">Mock modular forms and quantum modular forms</a>, Proc. Amer. Math. Soc. 144 (2016), 2337-2349. (See page 2341.)
%H J. H. Conway and N. J. A. Sloane, <a href="http://neilsloane.com/doc/Me150.pdf">Low-Dimensional Lattices. IV. The Mass Formula</a>, Proc. Roy. Soc. London Ser. A 419 (1988), no. 1857, 259-286. (See table 6.)
%H M. Monks, <a href="https://doi.org/10.1090/S0002-9939-09-10076-X">Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts</a>, Proc. Amer. Math. Soc. 138 (2010), no. 2, 481-494. (See page 485.)
%H D. Shanks and J. W. Wrench, <a href="https://doi.org/10.1090/S0025-5718-1963-0159796-4">The calculation of certain Dirichlet series</a>, Math. Comp. 17 (1963), 136-154. (See line 6 of Table 1.)
%F E.g.f.: cos(2*x) * cos(3*x) / cos(6*x).
%F From _Peter Luschny_, Apr 13 2022: (Start)
%F E.g.f.: (cos(x) + cos(5*x))*sec(6*x) / 2, even powers only.
%F a(n) = A000192(n)/2. (End)
%F 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
%p egf := (cos(x) + cos(5*x))*sec(6*x) / 2: ser := series(egf, x, 32):
%p seq(n!*coeff(ser, x ,n), n = 0..24, 2); # _Peter Luschny_, Apr 13 2022
%o (Sage)
%o x = PowerSeriesRing(QQ, 'x', default_prec=30).gen()
%o f = cos(2*x) * cos(3*x) / cos(6*x)
%o [cf for cf in f.egf_to_ogf() if cf]
%o (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
%Y Intermediate case between A002437 and A349429.
%Y Cf. A000192.
%K nonn,easy
%O 0,2
%A _F. Chapoton_, Apr 13 2022
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