Box k in row n shows the binary strings of length n with weight k.
Pascal's triangle (blue) counts pairs of symmetric strings as two, while Lozanić's triangle (green) counts them as one.
For n = 0 there is only the empty binary string.

Image set Numbers of binary strings and necklaces up to symmetry
strings
necklaces
all   =   sym. + chiral     chiral   =   all − sym.   symmetric   chiral up to reversal   =   chiral / 2   =   all u.t.r. − sym.   =   all − all u.t.r.   all up to reversal   =   sym. + chiral u.t.r.