how do you describe $a(n)$ with elementary functions?
$$a(n)=Ʒ_n(1)$$
$$Ʒ_n(x)=\intƷ_{n-1}(x) dx+\frac{d}{dx}Ʒ_{n-2}(x)$$ where $Ʒ_0(x)=1$ and $Ʒ_1(x)=1$
I found $Ʒ_n(x)$ with $n$ between $1$ and $7$
$$Ʒ_1(x)=1,Ʒ_2(x)=x,Ʒ_3(x)=\frac{x^2}{2},Ʒ_4(x)=\frac{x^3}{6}+1,Ʒ_5(x)=\frac{x^4}{24}+2x,Ʒ_6(x)=\frac{x^5}{120}+\frac{3x^2}{2},\small \dots$$
$$a(1)=1,a(2)=1,a(3)=\frac{1}{2},a(4)=1+\frac{1}{6},\dots$$