Enumeration of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus in the Centre

We compute the number of rhombus tilings of a hexagon with side lengths a, b, c, a, b, c which contain the central rhombus and the number of rhombus tilings of a hexagon with side lengths a, b, c, a, b, c which contain the “almost central” rhombus above the centre.