OFFSET
1,3
COMMENTS
The positions at which the signs of the terms change appear to be ~ c*n^2 (see example section).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..532
FORMULA
E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! may be defined by:
(1) A(x) = Integral exp(-x*tan(x)) / cos(x) dx.
(2) A(x) = lim_{N->oo} (x/N) * Sum_{n=1..N} cos((n+1)*x/N)^n / cos(n*x/N)^(n+1).
EXAMPLE
E.g.f.: A(x) = x - x^3/3! - 3*x^5/5! - 5*x^7/7! + 441*x^9/9! + 25911*x^11/11! + 1384757*x^13/13! + 74436531*x^15/15! + 3175224945*x^17/17! - 135369432209*x^19/19! + ...
where d/dx A(x) = exp(-x*tan(x)) / cos(x).
Also, e.g.f. A(x) equals the limit of the finite sum:
A(x) = lim_{N->oo} (x/N) * [1 + cos(2*x/N)/cos(x/N)^2 + cos(3*x/N)^2/cos(2*x/N)^3 + cos(4*x/N)^3/cos(3*x/N)^4 + cos(5*x/N)^4/cos(4*x/N)^5 + cos(6*x/N)^5/cos(5*x/N)^6 + ... + cos(x)^(N-1)/cos((N-1)*x/N)^N].
PATTERN OF SIGNS.
The signs (+-1) of the terms begin:
[+, -, -, -, +, +, +, +, +, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, ...].
The positions at which the signs of the terms change begin as follows:
[1, 2, 5, 10, 18, 29, 42, 57, 75, 95, 118, 143, 171, 201, 234, 269, 307, 347, 390, 435, 482, 532, 585, 639, 697, 757, 819, 884, 951, 1021, 1093, 1167, 1245, 1324, 1406, 1491, 1578, 1667, ..., A354246(n), ...]
which appears to be asymptotic to c*n^2 for some constant c ~ 1.2...
MATHEMATICA
nmax = 20; Table[(CoefficientList[Series[1/(E^(x*Tan[x])*Cos[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[k]], {k, 1, 2*nmax, 2}] (* Vaclav Kotesovec, May 24 2022 *)
PROG
(PARI) {a(n) = my(A = intformal( exp(-x * tan(x +O(x^(2*n+1))))/cos(x +O(x^(2*n+1)) ) )); (2*n-1)!*polcoeff(A, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, May 20 2022
STATUS
approved