Playing with Factorials

We take 4 consecutive integers such that

((1 !) + (2 !) + (3 !) + (4 !)) / (1 * 2 * 3 * 4) = 1.37500

((2 !) + (3 !) + (4 !) + (5 !)) / (2 * 3 * 4 * 5) = 1.26666667

((3 !) + (4 !) + (5 !) + (6 !)) / (3 * 4 * 5 * 6) = 2.41666667

((4 !) + (5 !) + (6 !) + (7 !)) / (4 * 5 * 6 * 7) = 7.02857143

((5 !) + (6 !) + (7 !) + (8 !)) / (5 * 6 * 7 * 8) = 27.5

((6 !) + (7 !) + (8 !) + (9 !)) / (6 * 7 * 8 * 9) = 135.238095

((7 !) + (8 !) + (9 !) + (10 !)) / (7 * 8 * 9 * 10) = 801

We get our first integer

If we continue onto the next series, we get

((8 !) + (9 !) + (10 !) + (11 !)) / (8 * 9 * 10 * 11) = 5549.09091

This is not an integer. Could you predict when we could get an integer?