Sum of squares of even combinatorial coefficients

Question

Find the value of $${100 \choose0}^2+{100 \choose2}^2+{100 \choose4}^2+\cdots+{100 \choose100}^2$$ I can calculate the sum of squares of all coefficients but not even coefficients.

Answer 1

The given expression is equal to the coefficient of $x^{100}$ in $$(\frac{1}{4})[(x+1)^{100}+(x-1)^{100}][(1+x)^{100}+(1-x)^{100}]$$

= coefficient of $ x^{100}$ in$ (\frac{1}{4})((1+x)^{100}+(1-x)^{100})^2$

$$=(\frac{1}{2}){200 \choose100} + (\frac{1}{2}){100 \choose50}$$