  The class B of lacunary polynomials f(x) := -1 + x + x^n + x^{m_1} + x^{m_2}
+ ... + x^{m_s}, where s >= 0, m_1 - n >= n - 1, m_{q+1} - m_{q} >= n - 1 for 1
<= q < s, n >= 3 is studied. A polynomial having its coefficients in {0, 1}
except its constant coefficient equal to -1 is called an almost Newman
polynomial. A general theorem of factorization of the almost Newman polynomials
of the class B is obtained. Such polynomials possess lenticular roots in the
open unit disk off the unit circle in the small angular sector \pi/18 <= arg z
<= \pi/18 and their nonreciprocal parts are always irreducible. The existence
of lenticuli of roots is a peculiarity of the class B. By comparison with the
Odlyzko - Poonen Conjecture and its variant Conjecture, an `Asymptotic
Reducibility Conjecture' is formulated aiming at establishing the proportion of
irreducible polynomials in this class. This proportion is conjectured to be 3/4
and estimated using Monte-Carlo methods. The numerical approximate value ~
0.756 is obtained. The results extend those on trinomials (Selmer) and
quadrinomials (Ljunggren, Mills, Finch and Jones).
