[seqfan] g.f. for a(n)= a(n-1)*2^n+n, a(0)=1 ?

[Georgi Guninski]

i tried to submit:
a(n)= a(n-1)*2^n+n, a(0)=1

it satisfies both:
a(n+1) = (2^(n + 1) + 1)*a(n) - 2^n*a(n - 1) + 1

a(n+1) = ((a(n - 2) + 4*a(n - 1) + 4)*a(n) - 2*a(n - 1)^2 - 4*a(n)^2 +
a(n - 2) - 4*a(n - 1))/(a(n - 2) - 2*a(n - 1))

and is kind of related to:
A010842 a(n) = n*a(n-1)+2^n

superseeker fails to find g.f., does anyone know one ?