  By using one of the definitions of the Bernoulli numbers, we prove that they
solve particular odd and even lower triangular Toeplitz (l.t.T.) systems of
equations. In a paper Ramanujan writes down a sparse lower triangular system
solved by Bernoulli numbers; we observe that such system is equivalent to a
sparse l.t.T. system. The attempt to obtain the sparse l.t.T. Ramanujan system
from the l.t.T. odd and even systems, has led us to study efficient methods for
solving generic l.t.T. systems. Such methods are here explained in detail in
case n, the number of equations, is a power of b, b=2,3 and b generic.
