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%I #20 Oct 10 2024 15:21:13
%S 3,4,4,4,4,5,4,4,4,4,5,4,4,5,4,4,4,4,5,4,4,5,4,4,5,4,4,4,4,5,4,4,5,4,
%T 4,4,4,5,4,4
%N Number of Maclaurin polynomials p(2m,x) of cos x that have exactly n positive zeros.
%C Maclaurin polynomial p(2m,x) is 1 - x^2/2! + x^4/4! + ... + (-1)^m x^(2m)/(2m)!.
%e a(1) counts these values of 2m: 2, 6, and 10. The single positive zeros of p(2,x), p(6,x), and p(10,x) are 1.41421..., 1.56990..., and 1.57079..., respectively.
%t z = 30; p[m_, x_] := Normal[Series[Cos[x], {x, 0, m }]];
%t t[n_] := x /. NSolve[p[n, x] == 0, x, z];
%t u[n_] := Select[t[n], Im[#] == 0 && # > 0 &];
%t v = Table[Length[u[n]], {n, 2, 100, 2}]
%t Table[Count[v, n], {n, 1, 10}]
%Y Cf. A332326, A332420.
%K nonn,more
%O 1,1
%A _Clark Kimberling_, Feb 11 2020