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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.

%I A354246 #40 Mar 22 2025 19:03:33
%S A354246 1,2,5,10,18,29,42,57,75,95,118,143,171,201,234,269,307,347,390,435,
%T A354246 482,532,585,639,697,757,819,884,951,1021,1093,1167,1245,1324,1406,
%U A354246 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
%N 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.
%C A354246 The e.g.f. of A354245 is Integral exp(-x*tan(x))/cos(x) dx.
%C A354246 What is the limit of a(n)/n^2 ?
%C A354246 Conjecture: lim_{n->oo} a(n)/n^2 = Pi^2/8 = A111003 = 1.2337... - _Vaclav Kotesovec_, May 26 2022
%e A354246 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)! + ...
%e A354246 The signs (+-1) of the coefficients A354245 begin:
%e A354246 [+, -, -, -, +, +, +, +, +, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, ...].
%e A354246 This sequence gives the positions in A354245 at which the signs of the coefficients change.
%t A354246 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 *)
%Y A354246 Cf. A354245, A354399, A354425.
%K A354246 nonn
%O A354246 1,2
%A A354246 _Paul D. Hanna_, May 20 2022
%E A354246 a(39)-a(64) from _Vaclav Kotesovec_, May 26 2022