Here the coefficients a(i) are defined inductively by
a(0)=1
a(i) = a(i-1)+(i+1)^2 for i>0
So then we can express the sum S as
S = a(0) + (a(0) + 2^2)x + (a(1) + 3^2)x^2 + (a(2) + 4^2)x^3 +...
= 1 + xS + 4x + 9x^2 +16x^3 +...
= xS + the old series
So we can solve for S; S= (1+x)/(1-x)^4