This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354245 #42 May 25 2022 02:21:28
%S A354245 1,-1,-3,-5,441,25911,1384757,74436531,3175224945,-135369432209,
%T A354245 -89771310955155,-25527579751884693,-6567045994040209879,
%U A354245 -1678101422880410465625,-427686430807976068014939,-102728760825086263958009309,-18156608776369804213731821343,2585946334251026101959272934111
%N A354245 E.g.f.: Integral exp(-x*tan(x)) / cos(x) dx.
%C A354245 The positions at which the signs of the terms change appear to be ~ c*n^2 (see example section).
%H A354245 Paul D. Hanna, <a href="/A354245/b354245.txt">Table of n, a(n) for n = 1..532</a>
%F A354245 E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! may be defined by:
%F A354245 (1) A(x) = Integral exp(-x*tan(x)) / cos(x) dx.
%F A354245 (2) A(x) = lim_{N->oo} (x/N) * Sum_{n=1..N} cos((n+1)*x/N)^n / cos(n*x/N)^(n+1).
%e A354245 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! + ...
%e A354245 where d/dx A(x) = exp(-x*tan(x)) / cos(x).
%e A354245 Also, e.g.f. A(x) equals the limit of the finite sum:
%e A354245 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].
%e A354245 PATTERN OF SIGNS.
%e A354245 The signs (+-1) of the terms begin:
%e A354245 [+, -, -, -, +, +, +, +, +, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, ...].
%e A354245 The positions at which the signs of the terms change begin as follows:
%e A354245 [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), ...]
%e A354245 which appears to be asymptotic to c*n^2 for some constant c ~ 1.2...
%t A354245 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 *)
%o A354245 (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)}
%o A354245 for(n=1,20,print1(a(n),", "))
%Y A354245 Cf. A009244, A009264, A009273, A354020, A354246.
%K A354245 sign
%O A354245 1,3
%A A354245 _Paul D. Hanna_, May 20 2022