Consider the following integral:
$2\pi\int\limits_0^\pi \sin(x) \cos^2(x) Y_{l,0}(x) Y_{l+2,0}(x) \mathrm{d}x$
Wherein $Y_{l,0}$ are the spherical harmonics for $m=0$, so they are not dependend of $\phi$, therefore the $2\pi$ in front of the integral ($\phi$ integration). Can the solution of this be written as a sequence? The solution will be:
$2/(3 \sqrt{5}), (2 \sqrt{3/7})/5, 4/(7 \sqrt{5}), 20/(9 \sqrt{77}), \ 10/(11 \sqrt{13}), (14 \sqrt{3/55})/13, 56/(15 \sqrt{221}),\dots$
for $l=0,1,2,\dots$
This should be sufficient hint: You want to compute $$\int_{-1}^{1}t^2P_l(t)P_{l+2}(t)dt.$$ Here $t=\cos x$, $P_l$'s are Legendre polynomials, and you need to multiply the normalization coefficients for $Y_l$'s and $2\pi$. Now use the formula $$tP_l(t)=((l+1)P_{l+1}(t)+(l-1)P_{l-1})/(2l+1)$$ to rewrite the integrant $(tP_l(t))(tP_{l+2})$ as a linear combination of products of $P_k$'s, and then use the orthogonality relation $$\int_{-1}^{1}P_m(t)P_n(t)dt=2\delta_{mn}/(2n+1)$$ (notice that only the term with $n=m=l+1$ survives).