### Creating symmetric polynomials of squares of variables?

I have five variables: a,b,c,d,e, for which I want to use the elementary symmetric polynomials of their squares:

s1 = a^2+b^2+c^2+d^2+e^2

s2 = a^2b^2 + a^2c^2 + ... + d^2e^2

and so on, to:

s5 = a^2b^2c^2d^2e^2.

Now, I can do this by hand, or by pulling out the coefficients of the polynomial

P = (x+a^2)(x+b^2)...(x+e^2)

However, is there a nicer way which uses Sage's own extensive symmetric polynomial functionality?