A332420 -id:A332420 - cp's OEIS frontend

A332325 Number of Maclaurin polynomials p(2m,x) of cos(x) that have exactly n positive zeros.

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, 4, 4, 4, 5, 4, 4, 5, 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, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4

Offset: 1

Clark Kimberling, Feb 11 2020

nonn

Maclaurin polynomial p(2m,x) of cos(x) is 1 - x^2/2! + x^4/4! - ... + (-1)^m*x^(2m)/(2m)!.

			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 sqrt(2), 1.56990..., and 1.57079..., respectively.

Cf. A332326, A332420, A358997.

z = 30; p[m_, x_] := Normal[Series[Cos[x], {x, 0, m }]];
t[n_] := x /. NSolve[p[n, x] == 0, x, z];
u[n_] := Select[t[n], Im[#] == 0 && # > 0 &];
v = Table[Length[u[n]], {n, 2, 100, 2}]
Table[Count[v, n], {n, 1, 10}]

More terms from Jinyuan Wang, Jan 21 2025

A383644 a(n) is the number of zeros in the left half-plane of the Maclaurin polynomial of degree n for exp(z).

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10, 11, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 16, 17, 18, 19, 20, 19, 20, 21, 22, 23, 24, 23, 24, 25, 26, 27, 26, 27, 28, 29, 30, 29, 30, 31, 32, 33, 32, 33, 34, 35, 36, 37, 36, 37, 38, 39, 40, 39, 40, 41, 42, 43, 42, 43, 44

Offset: 1

Michel Lagneau, May 03 2025

nonn

The Maclaurin polynomial of degree n for exp(z) is P(n,z) = Sum_{i=0..n} z^i/i!

The number of zeros in the right half-plane is equal to n - a(n) because we do not observe any purely imaginary roots.

			a(4)= 4 because P(4,z) = 1 + z/1! + z^2/2! + z^3/3! + z^4/4! with 4 roots in the left half-plane:
z1 = -1.729444231-.8889743761*i,
z2 = -1.729444231+.8889743761*i,
z3 = -.2705557689-2.504775904*i,
z4 = -.2705557689+2.504775904*i

Cf. A332324, A330187, A332420.

A:=proc(n) local P, m, y, it:
  it:=0:P:=add(x^i/i!,i=0..n):
   y:=[fsolve(expand(P), x, complex)]:
    for m from 1 to nops(y) do:
     if Re(y[m])<0 then
      it:=it+1:else fi:
    od: A(n):=it:end proc:
seq(A(n), n=1..70);

cp's OEIS Frontend

A332325 Number of Maclaurin polynomials p(2m,x) of cos(x) that have exactly n positive zeros.

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