A024179 [ (4th elementary symmetric function of 2,3,...,n+4)/(2+3+...+n+4) ].

8, 52, 189, 526, 1242, 2609, 5024, 9043, 15411, 25108, 39392, 59842, 88414, 127496, 179963, 249241, 339377, 455103, 601915, 786146, 1015050, 1296888, 1641014, 2057967, 2559573, 3159036, 3871051, 4711904, 5699589, 6853918, 8196644, 9751581