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

0,2

Comments

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

Links

Table of n, a(n) for n=0..11.

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

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)

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 *)

Prog

(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]

Crossrefs

Sequence in context: A266032 A269123 A336171 * A196429 A243779 A267335

Adjacent sequences: A349426 A349427 A349428 * A349430 A349431 A349432

Keyword

nonn

Author

F. Chapoton, Nov 17 2021

Status

approved