A shifted binomial theorem and trigonometric series

We introduce a shifted version of the binomial theorem, and use it to study some remarkable trigonometric integrals and their explicit rewriting in terms of binomial multiple sums. Motivated by the expressions of area generating functions arising in the counting of closed walks on various lattices, we propose similar sums involving fractional values of the area and show that they are closely related to their integer counterparts and lead to rational sequences converging to powers of $\pi$. Our results, other than their mathematical interest, could be relevant to generalizations of statistical mechanical models of the Heisenberg chain type involving higher spins or $SU(N)$ degrees of freedom.