A349429 Expansion of e.g.f. cos(5*x)*cos(9*x)/cos(15*x) (even powers only).

1, 119, 129361, 353851559, 1806970377121, 14829833979504599, 178506068100424343281, 2962559872323037509279239, 64836735740991992791046187841, 1809194806338763806974577192135479, 62691937652492245112191045131692230801, 2641170468091820745160358034750851940073319

Offset: 0

Author

F. Chapoton, Nov 17 2021

Keywords

• nonn

Comments

Only terms of even indices are given. See Formula (10) in the Lawrence-Zagier article.

Links

• Ruth Lawrence and Don Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), no. 1, 93-107.

Programs

Maple

A349429 := n -> (-900)^n*(euler(2*n, 1/30) + euler(2*n, 11/30)) / 2:
seq(A349429(n), n = 0..11); # Peter Luschny, Nov 17 2021

Mathematica

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 *)

Sage

x = PowerSeriesRing(QQ, 'x', default_prec=30).gen()
f = cos(5*x) * cos(9*x) / cos(15*x)
[cf for cf in f.egf_to_ogf() if cf]

Formula

E.g.f.: cos(5*x) * cos(9*x) / cos(15*x).

From Peter Luschny, Nov 17 2021: (Start)

a(n) = (-900)^n*(E(2*n, 1/30) + E(2*n, 11/30)) / 2, where E(n, x) are the Euler polynomials.

a(n) ~ c*(2*n)!*(30/Pi)^(2*n) where c = 0.64812598778325714671749857159... (End)