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
Keywords
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.
Programs
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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
Comments