Finding series coefficients

Given

b[0]:= 1; Sum[Binomial[n, k]*((2*n-2*k-1)!!)^2*b[k+1], {k,0,n}] = ((2*n+1)!!)^2; 

is there a way to find the coefficients b[n] using InverseSeries, SeriesCoefficient, or some other method?