This paper makes use of the recently introduced technique of γ-operators to evaluate the Hankel determinant with binomial coefficient entries ak = (3 k)!/ (2k)! k! . We actually evaluate the determinant of a class of polynomials ak(x) having this binomial coefficient as constant term. The evaluation in the polynomial case is as an almost product, i.e. as a sum of a small number of products. The γ-operator technique to find the explicit form of the almost product relies on differential-convolution equations and establishes a second order differential equation for the determinant. In addition to x=0, product form evaluations for x = &threefifths;, &threequarters;, &threehalves;, 3 are also presented. At x=1, we obtain another almost product evaluation for the Hankel determinant with ak = ( 3 k+1)!/(2k+1)!k!  .