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

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

Author

Michel Lagneau, May 03 2025

Keywords

• nonn

Comments

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.

Examples

			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
		

Crossrefs

Cf. A332324, A330187, A332420.

Programs

Maple

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);