Generating functions, sequences with unlimited history of recursion

Question

There is sequence $a_n = a_{n - 1} + 2 a_{n - 2} + \dots + n a_0$, and $a_0 = 1$. I found the generating function for this sequence: $(3t^2-3t+1)/(1-2t)^2$ but I do not know what to do next. How can I solve this? Should be reduced to a finite sum? Thank you in advance

Answer 1

You expand the generating function as 1+x+3*x^2+8*x^3+20*x^4+... and look the coefficients up in http://oeis.org/A001792 .