A009522 "Pascal sweep" for k=8: draw a horizontal line through the 1 at C(k,0) in Pascal's triangle; rotate this line and record the sum of the numbers on it (excluding the initial 1).

1, 9, 36, 84, 127, 10, 171, 1, 120, 11, 350, 12, 165, 1, 318, 13, 331, 220, 78, 14, 1, 1596, 1, 16, 105, 364, 716, 17, 912, 1, 455, 18, 1600, 19, 560, 1, 1440, 20, 1366, 680, 171, 21, 1, 5895, 1, 23, 210, 969, 2381, 24, 3234, 1, 1140, 25, 5030, 26, 1330, 1, 4644, 27, 3877