OFFSET
1,2
COMMENTS
The e.g.f. of A354245 is Integral exp(-x*tan(x))/cos(x) dx.
What is the limit of a(n)/n^2 ?
Conjecture: Lim_{n->infinity) a(n)/n^2 = Pi^2/8 = A111003 = 1.2337... - Vaclav Kotesovec, May 26 2022
EXAMPLE
The expansion of Integral exp(-x*tan(x)) / cos(x) dx = 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! + ... + A354245(n)*x^(2*n-1)/(2*n-1)! + ...
The signs (+-1) of the coefficients A354245 begin:
[+, -, -, -, +, +, +, +, +, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, ...].
This sequence gives the positions in A354245 at which the signs of the coefficients change.
MATHEMATICA
nmax = 500; A354245 = 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}]; Join[{1}, Select[Range[nmax], A354245[[#]]*A354245[[#-1]] < 0 &]] (* Vaclav Kotesovec, May 24 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 20 2022
EXTENSIONS
a(39)-a(64) from Vaclav Kotesovec, May 26 2022
STATUS
approved