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

%I A383644 #13 May 20 2025 16:53:24
%S A383644 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,
%T A383644 18,19,20,19,20,21,22,23,24,23,24,25,26,27,26,27,28,29,30,29,30,31,32,
%U A383644 33,32,33,34,35,36,37,36,37,38,39,40,39,40,41,42,43,42,43,44
%N A383644 a(n) is the number of zeros in the left half-plane of the Maclaurin polynomial of degree n for exp(z).
%C A383644 The Maclaurin polynomial of degree n for exp(z) is P(n,z) = Sum_{i=0..n} z^i/i!
%C A383644 The number of zeros in the right half-plane is equal to n - a(n) because we do not observe any purely imaginary roots.
%e A383644 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:
%e A383644 z1 = -1.729444231-.8889743761*i,
%e A383644 z2 = -1.729444231+.8889743761*i,
%e A383644 z3 = -.2705557689-2.504775904*i,
%e A383644 z4 = -.2705557689+2.504775904*i
%p A383644 A:=proc(n) local P, m, y, it:
%p A383644   it:=0:P:=add(x^i/i!,i=0..n):
%p A383644    y:=[fsolve(expand(P), x, complex)]:
%p A383644     for m from 1 to nops(y) do:
%p A383644      if Re(y[m])<0 then
%p A383644       it:=it+1:else fi:
%p A383644     od: A(n):=it:end proc:
%p A383644 seq(A(n), n=1..70);
%Y A383644 Cf. A332324, A330187, A332420.
%K A383644 nonn
%O A383644 1,2
%A A383644 _Michel Lagneau_, May 03 2025