A354246 Indices of coefficients of x^(2*k-1) in Integral exp(-x*tan(x))/cos(x) dx at which the signs of the coefficients change: list of k such that sign(A354245(k)) != sign(A354245(k-1)), starting with 1.

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, 1759, 1853, 1950, 2050, 2151, 2256, 2362, 2471, 2583, 2697, 2814, 2933, 3054, 3178, 3305, 3434, 3565, 3699, 3835, 3974, 4115, 4259, 4405, 4554, 4705, 4859

Offset: 1

Author

Paul D. Hanna, May 20 2022

Keywords

• nonn

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->oo} a(n)/n^2 = Pi^2/8 = A111003 = 1.2337... - Vaclav Kotesovec, May 26 2022

Examples

			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.
		

Crossrefs

Cf. A354245, A354399, A354425.

Programs

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 *)

Extensions

a(39)-a(64) from Vaclav Kotesovec, May 26 2022