|
%I #25 Nov 18 2021 12:41:39
%S 1,119,129361,353851559,1806970377121,14829833979504599,
%T 178506068100424343281,2962559872323037509279239,
%U 64836735740991992791046187841,1809194806338763806974577192135479,62691937652492245112191045131692230801,2641170468091820745160358034750851940073319
%N Expansion of e.g.f. cos(5*x)*cos(9*x)/cos(15*x) (even powers only).
%C Only terms of even indices are given. See Formula (10) in the Lawrence-Zagier article.
%H Ruth Lawrence and Don Zagier, <a href="https://doi.org/10.4310/AJM.1999.v3.n1.a5">Modular forms and quantum invariants of 3-manifolds</a>, Asian J. Math. 3 (1999), no. 1, 93-107.
%F E.g.f.: cos(5*x) * cos(9*x) / cos(15*x).
%F From _Peter Luschny_, Nov 17 2021: (Start)
%F a(n) = (-900)^n*(E(2*n, 1/30) + E(2*n, 11/30)) / 2, where E(n, x) are the Euler polynomials.
%F a(n) ~ c*(2*n)!*(30/Pi)^(2*n) where c = 0.64812598778325714671749857159... (End)
%p A349429 := n -> (-900)^n*(euler(2*n, 1/30) + euler(2*n, 11/30)) / 2:
%p seq(A349429(n), n = 0..11); # _Peter Luschny_, Nov 17 2021
%t m = 13; Take[CoefficientList[Series[Cos[5*x]*Cos[9*x]/Cos[15*x], {x, 0, 2*m}], x] * Range[0, 2*m]!, {1, 2*m + 1, 2}] (* _Amiram Eldar_, Nov 17 2021 *)
%o (Sage)
%o x = PowerSeriesRing(QQ, 'x', default_prec=30).gen()
%o f = cos(5*x) * cos(9*x) / cos(15*x)
%o [cf for cf in f.egf_to_ogf() if cf]
%K nonn
%O 0,2
%A _F. Chapoton_, Nov 17 2021
| 