The list of length 21 whose i-th entry is the enumerating function for the area under Dyck paths of length i-1 is [ 4 2 9 7 5 3 [1, 0, q, 0, q + q , 0, q + q + 2 q + q , 0, 16 14 12 10 8 6 4 25 23 q + q + 2 q + 3 q + 3 q + 3 q + q , 0, q + q 21 19 17 15 13 11 9 7 + 2 q + 3 q + 5 q + 5 q + 7 q + 7 q + 6 q + 4 q 5 36 34 32 30 28 26 24 + q , 0, q + q + 2 q + 3 q + 5 q + 7 q + 9 q 22 20 18 16 14 12 10 + 11 q + 14 q + 16 q + 16 q + 17 q + 14 q + 10 q 8 6 49 47 45 43 41 39 + 5 q + q , 0, q + q + 2 q + 3 q + 5 q + 7 q 37 35 33 31 29 27 25 + 11 q + 13 q + 18 q + 22 q + 28 q + 32 q + 37 q 23 21 19 17 15 13 11 + 40 q + 44 q + 43 q + 40 q + 35 q + 25 q + 15 q 9 7 64 62 60 58 56 54 + 6 q + q , 0, q + q + 2 q + 3 q + 5 q + 7 q 52 50 48 46 44 42 40 + 11 q + 15 q + 20 q + 26 q + 34 q + 42 q + 53 q 38 36 34 32 30 28 + 63 q + 73 q + 85 q + 96 q + 106 q + 113 q 26 24 22 20 18 16 + 118 q + 118 q + 115 q + 102 q + 86 q + 65 q 14 12 10 8 81 79 77 75 + 41 q + 21 q + 7 q + q , 0, q + q + 2 q + 3 q 73 71 69 67 65 63 61 + 5 q + 7 q + 11 q + 15 q + 22 q + 28 q + 38 q 59 57 55 53 51 49 + 48 q + 63 q + 77 q + 97 q + 116 q + 139 q 47 45 43 41 39 37 + 162 q + 190 q + 215 q + 245 q + 268 q + 293 q 35 33 31 29 27 25 + 314 q + 331 q + 338 q + 338 q + 326 q + 303 q 23 21 19 17 15 13 + 268 q + 219 q + 167 q + 112 q + 63 q + 28 q 11 9 100 98 96 94 92 90 + 8 q + q , 0, q + q + 2 q + 3 q + 5 q + 7 q 88 86 84 82 80 78 76 + 11 q + 15 q + 22 q + 30 q + 40 q + 52 q + 69 q 74 72 70 68 66 64 + 87 q + 111 q + 138 q + 171 q + 207 q + 249 q 62 60 58 56 54 52 + 295 q + 348 q + 405 q + 466 q + 531 q + 598 q 50 48 46 44 42 40 + 665 q + 734 q + 801 q + 862 q + 918 q + 958 q 38 36 34 32 30 28 + 990 q + 1003 q + 995 q + 959 q + 901 q + 813 q 26 24 22 20 18 16 + 704 q + 574 q + 434 q + 301 q + 182 q + 92 q 14 12 10] + 36 q + 9 q + q ] Note: this gives us the following enumerating function for areas under Dyck paths of length 0 through 20 2 / 4 2\ 4 / 9 7 5 3\ 6 d := 1 + x q + \q + q / x + \q + q + 2 q + q / x / 16 14 12 10 8 6 4\ 8 / 25 + \q + q + 2 q + 3 q + 3 q + 3 q + q / x + \q 23 21 19 17 15 13 11 9 + q + 2 q + 3 q + 5 q + 5 q + 7 q + 7 q + 6 q 7 5\ 10 / 36 34 32 30 28 26 + 4 q + q / x + \q + q + 2 q + 3 q + 5 q + 7 q 24 22 20 18 16 14 12 + 9 q + 11 q + 14 q + 16 q + 16 q + 17 q + 14 q 10 8 6\ 12 / 49 47 45 43 41 + 10 q + 5 q + q / x + \q + q + 2 q + 3 q + 5 q 39 37 35 33 31 29 27 + 7 q + 11 q + 13 q + 18 q + 22 q + 28 q + 32 q 25 23 21 19 17 15 13 + 37 q + 40 q + 44 q + 43 q + 40 q + 35 q + 25 q 11 9 7\ 14 / 64 62 60 58 56 + 15 q + 6 q + q / x + \q + q + 2 q + 3 q + 5 q 54 52 50 48 46 44 42 + 7 q + 11 q + 15 q + 20 q + 26 q + 34 q + 42 q 40 38 36 34 32 30 + 53 q + 63 q + 73 q + 85 q + 96 q + 106 q 28 26 24 22 20 18 + 113 q + 118 q + 118 q + 115 q + 102 q + 86 q 16 14 12 10 8\ 16 / 81 79 + 65 q + 41 q + 21 q + 7 q + q / x + \q + q 77 75 73 71 69 67 65 + 2 q + 3 q + 5 q + 7 q + 11 q + 15 q + 22 q 63 61 59 57 55 53 + 28 q + 38 q + 48 q + 63 q + 77 q + 97 q 51 49 47 45 43 41 + 116 q + 139 q + 162 q + 190 q + 215 q + 245 q 39 37 35 33 31 29 + 268 q + 293 q + 314 q + 331 q + 338 q + 338 q 27 25 23 21 19 17 + 326 q + 303 q + 268 q + 219 q + 167 q + 112 q 15 13 11 9\ 18 / 100 98 96 + 63 q + 28 q + 8 q + q / x + \q + q + 2 q 94 92 90 88 86 84 82 + 3 q + 5 q + 7 q + 11 q + 15 q + 22 q + 30 q 80 78 76 74 72 70 + 40 q + 52 q + 69 q + 87 q + 111 q + 138 q 68 66 64 62 60 58 + 171 q + 207 q + 249 q + 295 q + 348 q + 405 q 56 54 52 50 48 46 + 466 q + 531 q + 598 q + 665 q + 734 q + 801 q 44 42 40 38 36 34 + 862 q + 918 q + 958 q + 990 q + 1003 q + 995 q 32 30 28 26 24 22 + 959 q + 901 q + 813 q + 704 q + 574 q + 434 q 20 18 16 14 12 10\ 20 + 301 q + 182 q + 92 q + 36 q + 9 q + q / x ---------------------------- This can be used to test the enumerating function for the areas under Dyck Paths of length 0 through K given by SeqF1 2 / 4 2\ 4 / 9 7 5 3\ 6 1 + x q + \q + q / x + \q + q + 2 q + q / x / 16 14 12 10 8 6 4\ 8 / 25 + \q + q + 2 q + 3 q + 3 q + 3 q + q / x + \q 23 21 19 17 15 13 11 9 + q + 2 q + 3 q + 5 q + 5 q + 7 q + 7 q + 6 q 7 5\ 10 / 36 34 32 30 28 26 + 4 q + q / x + \q + q + 2 q + 3 q + 5 q + 7 q 24 22 20 18 16 14 12 + 9 q + 11 q + 14 q + 16 q + 16 q + 17 q + 14 q 10 8 6\ 12 / 49 47 45 43 41 + 10 q + 5 q + q / x + \q + q + 2 q + 3 q + 5 q 39 37 35 33 31 29 27 + 7 q + 11 q + 13 q + 18 q + 22 q + 28 q + 32 q 25 23 21 19 17 15 13 + 37 q + 40 q + 44 q + 43 q + 40 q + 35 q + 25 q 11 9 7\ 14 / 64 62 60 58 56 + 15 q + 6 q + q / x + \q + q + 2 q + 3 q + 5 q 54 52 50 48 46 44 42 + 7 q + 11 q + 15 q + 20 q + 26 q + 34 q + 42 q 40 38 36 34 32 30 + 53 q + 63 q + 73 q + 85 q + 96 q + 106 q 28 26 24 22 20 18 + 113 q + 118 q + 118 q + 115 q + 102 q + 86 q 16 14 12 10 8\ 16 / 81 79 + 65 q + 41 q + 21 q + 7 q + q / x + \q + q 77 75 73 71 69 67 65 + 2 q + 3 q + 5 q + 7 q + 11 q + 15 q + 22 q 63 61 59 57 55 53 + 28 q + 38 q + 48 q + 63 q + 77 q + 97 q 51 49 47 45 43 41 + 116 q + 139 q + 162 q + 190 q + 215 q + 245 q 39 37 35 33 31 29 + 268 q + 293 q + 314 q + 331 q + 338 q + 338 q 27 25 23 21 19 17 + 326 q + 303 q + 268 q + 219 q + 167 q + 112 q 15 13 11 9\ 18 / 100 98 96 + 63 q + 28 q + 8 q + q / x + \q + q + 2 q 94 92 90 88 86 84 82 + 3 q + 5 q + 7 q + 11 q + 15 q + 22 q + 30 q 80 78 76 74 72 70 + 40 q + 52 q + 69 q + 87 q + 111 q + 138 q 68 66 64 62 60 58 + 171 q + 207 q + 249 q + 295 q + 348 q + 405 q 56 54 52 50 48 46 + 466 q + 531 q + 598 q + 665 q + 734 q + 801 q 44 42 40 38 36 34 + 862 q + 918 q + 958 q + 990 q + 1003 q + 995 q 32 30 28 26 24 22 + 959 q + 901 q + 813 q + 704 q + 574 q + 434 q 20 18 16 14 12 10\ 20 + 301 q + 182 q + 92 q + 36 q + 9 q + q / x ---------------------------- ---------------------------- It can also be used to test the k-th derivative with respect to q of d(x,q), where d(x,q) is the bivariate weight enumerator for Dyck paths of length n and area m, evaluated at q=1 DyckDer:=DerK(1,0,x^2*q,q,x,15,f) ---------------------------- For example: The Maclaurin polynomial of order 20 of the seventh entry of the list DyckDer should be the Maclaurin polynomial of order 20 of the sixth derivative with respect to q of d(x,q) evaluated at q=1 The Maclaurin polynomial of order 20 of the seventh entry of the list DyckDer is 20 18 16 243201543177600 x + 24376700200320 x + 2157804707760 x 14 12 10 + 162048664800 x + 9682446240 x + 410870880 x 8 6 + 9774720 x + 65520 x which is indeed equal to the 6th derivative of d evaluated at q=1 20 18 16 243201543177600 x + 24376700200320 x + 2157804707760 x 14 12 10 + 162048664800 x + 9682446240 x + 410870880 x 8 6 + 9774720 x + 65520 x