Theorem: define the Abel-sum type sequence by n ----- \ A[n](r, s) = ) / ----- k = 0 (k - 1 + p) (n - k + q) k binomial(n, k) binomial(n + k, k) (r + k) (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)*binomial(n+k,k)*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x^k,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 2 3 2 2 (n p - n p r + n p s - n r s - p + p r - p s + p r s - 3 n p + n r - 2 n s 2 + 5 p - 3 p r + 4 p s - 2 r s + 2 n - 8 p + 2 r - 4 s + 4) A[n](r, s)/(%1 2 2 2 2 2 2 ) - (n p + 2 n p s - n r - 2 n r s - 2 p r - p s + 2 p r + r s 2 2 - 3 n p - 3 n s + p + 6 p r + 5 p s - 4 r - 2 r s + 2 n - 3 p - 4 r /d \ 2 2 - 5 s + 2) |-- A[n](r, s)|/(%1) + (n p r + n p s - n r - n r s - p r \dr / / 2 \ 3 2 |d | - p r s + r + r s - n r - n s + p r + p s - r - s) |--- A[n](r, s)|/(%1) | 2 | \dr / 2 (n p - n r + p - p r - 2 n - 2 p) A[n + 1](r, s) /d \ - ------------------------------------------------- + |-- A[n + 1](r, s)| %1 \dr / / 2 \ 2 |d | (n p - n r + p r - r - n + p - 2 r - 1) |--- A[n + 1](r, s)| | 2 | \dr / - ------------------------------------------------------------- = 0 %1 2 2 %1 := 2 n p - 2 n r + p - r - 3 n - p - 2 r - 1 3 2 2 2 2 3 2 2 - (2 n + 5 n q - 4 n s + 4 n q - 6 n q s + 2 n s + q - 2 q s + q s 2 2 2 + 5 n + 8 n q - 6 n s + 3 q - 4 q s + s + 4 n + 3 q - 2 s + 1) 3 2 2 2 2 2 2 3 A[n](r, s)/(%1) + (n + 2 n q - n s + n q - n s + q s - 2 q s + s 2 2 2 /d \ + 3 n + 4 n q - 2 n s + q - s + 3 n + 2 q - s + 1) |-- A[n](r, s)|/(%1) \ds / q (n + q - s) A[n + 1](r, s) /d \ + ---------------------------- + |-- A[n + 1](r, s)| %1 \ds / / 2 \ 2 2 |d | (n + n q - 2 n s - q s + s + 2 n + q - 2 s + 1) |--- A[n + 1](r, s)| | 2 | \ds / - ---------------------------------------------------------------------- %1 = 0 2 2 2 %1 := n - 2 n s - q + s + 2 n + q - 2 s + 1 and in Maple notation (n*p^2-n*p*r+n*p*s-n*r*s-p^3+p^2*r-p^2*s+p*r*s-3*n*p+n*r-2*n*s+5*p^2-3*p*r+4*p* s-2*r*s+2*n-8*p+2*r-4*s+4)/(2*n*p-2*n*r+p^2-r^2-3*n-p-2*r-1)*A[n](r,s)-(n*p^2+2 *n*p*s-n*r^2-2*n*r*s-2*p^2*r-p^2*s+2*p*r^2+r^2*s-3*n*p-3*n*s+p^2+6*p*r+5*p*s-4* r^2-2*r*s+2*n-3*p-4*r-5*s+2)/(2*n*p-2*n*r+p^2-r^2-3*n-p-2*r-1)*diff(A[n](r,s),r )+(n*p*r+n*p*s-n*r^2-n*r*s-p*r^2-p*r*s+r^3+r^2*s-n*r-n*s+p*r+p*s-r-s)/(2*n*p-2* n*r+p^2-r^2-3*n-p-2*r-1)*diff(diff(A[n](r,s),r),r)-(n*p-n*r+p^2-p*r-2*n-2*p)/(2 *n*p-2*n*r+p^2-r^2-3*n-p-2*r-1)*A[n+1](r,s)+diff(A[n+1](r,s),r)-(n*p-n*r+p*r-r^ 2-n+p-2*r-1)/(2*n*p-2*n*r+p^2-r^2-3*n-p-2*r-1)*diff(diff(A[n+1](r,s),r),r) = 0 -(2*n^3+5*n^2*q-4*n^2*s+4*n*q^2-6*n*q*s+2*n*s^2+q^3-2*q^2*s+q*s^2+5*n^2+8*n*q-6 *n*s+3*q^2-4*q*s+s^2+4*n+3*q-2*s+1)/(n^2-2*n*s-q^2+s^2+2*n+q-2*s+1)*A[n](r,s)+( n^3+2*n^2*q-n^2*s+n*q^2-n*s^2+q^2*s-2*q*s^2+s^3+3*n^2+4*n*q-2*n*s+q^2-s^2+3*n+2 *q-s+1)/(n^2-2*n*s-q^2+s^2+2*n+q-2*s+1)*diff(A[n](r,s),s)+q*(n+q-s)/(n^2-2*n*s- q^2+s^2+2*n+q-2*s+1)*A[n+1](r,s)+diff(A[n+1](r,s),s)-(n^2+n*q-2*n*s-q*s+s^2+2*n +q-2*s+1)/(n^2-2*n*s-q^2+s^2+2*n+q-2*s+1)*diff(diff(A[n+1](r,s),s),s) = 0 ------------------------------------------------- This took, 0.196, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ A[n](r, s) = ) / ----- k = 0 2 (k - 1 + p) (n - k + q) k binomial(n, k) binomial(n + k, k) (r + k) (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)*binomial(n+k,k)^2*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x^k ,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 2 4 2 3 2 3 2 2 2 2 2 2 3 - (n p - 3 n p r + n p s + 3 n p r - 3 n p r s - n p r 2 2 2 3 5 4 4 3 2 + 3 n p r s - n r s - 2 n p + 6 n p r - 2 n p s - 6 n p r 3 2 3 2 2 3 6 5 5 + 6 n p r s + 2 n p r - 6 n p r s + 2 n p r s + p - 3 p r + p s 4 2 4 3 3 3 2 2 3 2 3 2 2 + 3 p r - 3 p r s - p r + 3 p r s - p r s - 8 n p + 17 n p r 2 2 2 2 2 2 3 2 2 4 - 7 n p s - 10 n p r + 14 n p r s + n r - 7 n r s + 20 n p 3 3 2 2 2 3 - 46 n p r + 18 n p s + 32 n p r - 40 n p r s - 6 n p r 2 3 5 4 4 3 2 + 26 n p r s - 4 n r s - 12 p + 29 p r - 11 p s - 22 p r 3 2 3 2 2 3 2 2 2 + 26 p r s + 5 p r - 19 p r s + 4 p r s + 23 n p - 30 n p r 2 2 2 2 3 2 2 + 16 n p s + 7 n r - 16 n r s - 78 n p + 128 n p r - 60 n p s 2 3 2 4 3 3 - 54 n p r + 88 n p r s + 4 n r - 28 n r s + 59 p - 110 p r + 48 p s 2 2 2 3 2 3 2 2 + 59 p r - 84 p r s - 8 p r + 40 p r s - 4 r s - 28 n p + 16 n r 2 2 2 3 - 12 n s + 148 n p - 152 n p r + 88 n p s + 28 n r - 64 n r s - 152 p 2 2 2 3 2 2 + 204 p r - 104 p s - 68 p r + 120 p r s + 4 r - 28 r s + 12 n 2 2 - 136 n p + 64 n r - 48 n s + 216 p - 184 p r + 112 p s + 28 r - 64 r s + 48 n - 160 p + 64 r - 48 s + 48) A[n](r, s)/((n p - n r - n + p - r - 1) 2 4 2 3 2 3 2 2 2 2 2 2 3 %1) + (2 n p - 5 n p r + 3 n p s + 3 n p r - 9 n p r s + n p r 2 2 2 4 2 3 5 4 4 + 9 n p r s - n r - 3 n r s - 2 n p + 2 n p r - 4 n p s 3 2 3 2 3 2 2 4 + 6 n p r + 10 n p r s - 10 n p r - 6 n p r s + 4 n p r 3 4 5 5 4 2 4 3 3 - 2 n p r s + 2 n r s + 3 p r + p s - 9 p r - p r s + 9 p r 3 2 2 4 2 3 4 2 3 2 2 - 3 p r s - 3 p r + 5 p r s - 2 p r s - 15 n p + 27 n p r 2 2 2 2 2 2 3 2 2 4 - 18 n p s - 9 n p r + 36 n p r s - 3 n r - 18 n r s + 22 n p 3 3 2 2 2 3 - 22 n p r + 36 n p s - 30 n p r - 72 n p r s + 38 n p r 2 4 5 4 4 3 2 3 + 36 n p r s - 8 n r - 2 p - 25 p r - 13 p s + 69 p r + 16 p r s 2 3 2 2 4 3 4 2 2 - 55 p r + 12 p r s + 13 p r - 20 p r s + 5 r s + 40 n p 2 2 2 2 2 3 2 - 46 n p r + 34 n p s + 6 n r - 34 n r s - 92 n p + 80 n p r 2 2 3 2 4 - 116 n p s + 48 n p r + 164 n p r s - 36 n r - 48 n r s + 20 p 3 3 2 2 2 3 2 4 + 76 p r + 64 p s - 192 p r - 76 p r s + 110 p r - 6 p r s - 14 r 3 2 2 2 2 + 18 r s - 45 n p + 24 n r - 21 n s + 182 n p - 116 n p r + 158 n p s 2 3 2 2 2 - 24 n r - 116 n r s - 77 p - 100 p r - 149 p s + 228 p r + 140 p r s 3 2 2 2 - 72 r - 12 r s + 18 n - 170 n p + 56 n r - 78 n s + 142 p + 50 p r 2 + 164 p s - 96 r - 86 r s + 60 n - 125 p - 4 r - 69 s + 42) /d \ 2 3 2 2 |-- A[n](r, s)|/((n p - n r - n + p - r - 1) %1) - (n p + 3 n p s \dr / 2 2 2 2 3 2 2 3 3 - 3 n p r - 6 n p r s + 2 n r + 3 n r s - 4 n p r - 2 n p s 2 2 2 4 3 3 2 3 + 6 n p r + 6 n p r s - 2 n r - 4 n r s + 3 p r + 2 p r s 2 3 2 2 4 4 2 2 2 2 2 - 6 p r - 3 p r s + 3 p r + r s - 6 n p - 12 n p s + 6 n r 2 3 2 2 2 + 12 n r s + 2 n p + 24 n p r + 18 n p s - 30 n p r - 12 n p r s 3 2 3 3 2 2 2 3 + 4 n r - 6 n r s - 4 p r - 2 p s - 12 p r - 12 p r s + 24 p r 2 4 3 2 2 2 + 18 p r s - 8 r - 4 r s + 11 n p + 11 n s - 12 n p - 44 n p r 2 3 2 2 2 - 46 n p s + 34 n r + 24 n r s + p + 24 p r + 15 p s + 6 p r 3 2 2 2 + 16 p r s - 20 r - 20 r s - 6 n + 22 n p + 24 n r + 34 n s - 6 p 2 - 44 p r - 34 p s + 10 r - 12 n + 11 p + 24 r + 23 s - 6) / 2 \ |d | 2 2 2 2 2 2 2 |--- A[n](r, s)|/((n + 1) %1) + (n p r + n p s - 2 n p r - 2 n p r s | 2 | \dr / 2 3 2 2 2 2 2 3 2 + n r + n r s - 2 n p r - 2 n p r s + 4 n p r + 4 n p r s 4 3 2 3 2 2 4 3 5 4 - 2 n r - 2 n r s + p r + p r s - 2 p r - 2 p r s + r + r s 2 2 2 2 2 2 2 - 3 n p r - 3 n p s + 3 n r + 3 n r s + 2 n p r + 2 n p s 2 3 2 2 2 2 3 + 2 n p r + 2 n p r s - 4 n r - 4 n r s - 2 p r - 2 p r s + p r 2 4 3 2 2 2 + p r s + r + r s + 2 n r + 2 n s - 6 n p r - 6 n p s + 2 n r 2 2 2 3 2 + 2 n r s + p r + p s + 4 p r + 4 p r s - 3 r - 3 r s + 4 n r + 4 n s / 3 \ 2 |d | - 3 p r - 3 p s - r - r s + 2 r + 2 s) |--- A[n](r, s)|/((n + 1) %1) + ( | 3 | \dr / 3 2 2 3 4 3 2 2 3 2 n p - 3 n p r + 3 n p r - n r + p - 3 p r + 3 p r - p r - 7 n p 2 3 2 2 2 + 14 n p r - 7 n r - 7 p + 14 p r - 7 p r + 16 n p - 16 n r + 16 p 3 - 16 p r - 12 n - 12 p) A[n + 1](r, s)/(%1 (p - r - 1)) - (3 n p 2 2 3 4 3 2 2 3 4 - 9 n p r + 9 n p r - 3 n r + 2 p - 5 p r + 3 p r + p r - r 2 2 3 2 3 - 18 n p + 36 n p r - 18 n r - 12 p + 18 p r - 6 r + 34 n p - 34 n r 2 2 /d \ + 22 p - 10 p r - 12 r - 21 n - 11 p - 10 r - 3) |-- A[n + 1](r, s)|/(%1 \dr / / 2 \ |d | 2 2 2 (p - r - 1)) + |--- A[n + 1](r, s)| - (n p - 2 n p r + n r + p r | 2 | \dr / 2 3 2 2 - 2 p r + r - 3 n p + 3 n r + p - 5 p r + 4 r + 2 n - 3 p + 5 r + 2) / 3 \ |d | |--- A[n + 1](r, s)|/(%1) = 0 | 3 | \dr / 2 2 3 2 3 2 %1 := 3 n p - 6 n p r + 3 n r + p - 3 p r + 2 r - 12 n p + 12 n r - 3 p 2 - 6 p r + 9 r + 11 n - p + 12 r + 5 5 4 4 3 2 3 3 2 2 3 - (4 n + 16 n q - 12 n s + 25 n q - 36 n q s + 12 n s + 19 n q 2 2 2 2 2 3 4 3 2 2 - 39 n q s + 24 n q s - 4 n s + 7 n q - 18 n q s + 15 n q s 3 5 4 3 2 2 3 4 3 3 - 4 n q s + q - 3 q s + 3 q s - q s + 16 n + 50 n q - 36 n s 2 2 2 2 2 3 2 2 + 57 n q - 78 n q s + 24 n s + 28 n q - 54 n q s + 30 n q s 3 4 3 2 2 3 3 2 2 - 4 n s + 5 q - 12 q s + 9 q s - 2 q s + 25 n + 57 n q - 39 n s 2 2 3 2 2 3 2 + 42 n q - 54 n q s + 15 n s + 10 q - 18 q s + 9 q s - s + 19 n 2 2 + 28 n q - 18 n s + 10 q - 12 q s + 3 s + 7 n + 5 q - 3 s + 1) 6 5 5 4 2 4 A[n](r, s)/((n + 1) %2) + (4 n + 18 n q - 12 n s + 32 n q - 38 n q s 4 2 3 3 3 2 3 2 3 3 2 4 + 8 n s + 28 n q - 40 n q s + 4 n q s + 8 n s + 12 n q 2 3 2 2 2 2 3 2 4 5 4 - 12 n q s - 24 n q s + 36 n q s - 12 n s + 2 n q + 4 n q s 3 2 2 3 4 5 5 4 2 - 28 n q s + 40 n q s - 22 n q s + 4 n s + 2 q s - 8 q s 3 3 2 4 5 5 4 4 3 2 + 12 q s - 8 q s + 2 q s + 15 n + 57 n q - 39 n s + 83 n q 3 3 2 2 3 2 2 2 2 2 3 - 104 n q s + 26 n s + 57 n q - 93 n q s + 30 n q s + 6 n s 4 3 2 2 3 4 5 4 + 18 n q - 30 n q s - 3 n q s + 24 n q s - 9 n s + 2 q - 2 q s 3 2 2 3 4 5 4 3 3 2 2 - 7 q s + 13 q s - 7 q s + s + 19 n + 60 n q - 44 n s + 69 n q 2 2 2 3 2 2 3 4 - 96 n q s + 30 n s + 34 n q - 66 n q s + 36 n q s - 4 n s + 6 q 3 2 2 4 3 2 2 2 - 14 q s + 9 q s - s + 6 n + 18 n q - 18 n s + 17 n q - 32 n q s 2 3 2 2 3 2 2 + 14 n s + 5 q - 13 q s + 10 q s - 2 s - 6 n - 6 n q - q - 2 q s 2 /d \ 6 5 + 2 s - 5 n - 3 q + s - 1) |-- A[n](r, s)|/((n + 1) %1) - (n + 4 n q \ds / 5 4 2 4 4 2 3 3 3 2 3 3 - 2 n s + 6 n q - 4 n q s - n s + 4 n q - 8 n q s + 4 n s 2 4 2 3 2 2 2 2 3 2 4 4 + n q + 4 n q s - 12 n q s + 8 n q s - n s + 2 n q s 3 2 4 5 4 2 3 3 2 4 5 6 - 4 n q s + 4 n q s - 2 n s + q s - 4 q s + 6 q s - 4 q s + s 5 4 4 3 2 3 3 2 2 3 + 5 n + 17 n q - 9 n s + 21 n q - 16 n q s - 2 n s + 11 n q 2 2 2 2 2 3 4 3 2 2 - 3 n q s - 18 n q s + 10 n s + 2 n q + 6 n q s - 21 n q s 3 4 4 3 2 2 3 4 5 4 + 16 n q s - 3 n s + 2 q s - 5 q s + 3 q s + q s - s + 10 n 3 3 2 2 2 3 2 + 28 n q - 16 n s + 27 n q - 24 n q s + 10 n q - 6 n q s 2 3 4 3 2 2 3 4 3 - 12 n q s + 8 n s + q + 2 q s - 9 q s + 8 q s - 2 s + 10 n 2 2 2 2 3 2 2 + 22 n q - 14 n s + 15 n q - 16 n q s + 2 n s + 3 q - 3 q s - 2 q s 3 2 2 2 + 2 s + 5 n + 8 n q - 6 n s + 3 q - 4 q s + s + n + q - s) / 2 \ |d | / 2 |--- A[n](r, s)| / ((n + n q - n s + q - s - 1) %2) | 2 | / \ds / 2 2 2 (n + 2 n q - 2 n s + q - 2 q s + s ) q A[n + 1](r, s) + ------------------------------------------------------- %2 /d \ 4 3 3 2 2 2 + |-- A[n + 1](r, s)| - (2 n + 5 n q - 8 n s + 3 n q - 15 n q s \ds / 2 2 3 2 2 3 4 3 2 2 + 12 n s - n q - 6 n q s + 15 n q s - 8 n s - q + q s + 3 q s 3 4 3 2 2 2 2 - 5 q s + 2 s + 5 n + 12 n q - 15 n s + 9 n q - 24 n q s + 15 n s 3 2 2 3 2 2 + 2 q - 9 q s + 12 q s - 5 s + 3 n + 7 n q - 6 n s + 4 q - 7 q s / 2 \ 2 |d | 4 3 3 + 3 s - n + s - 1) |--- A[n + 1](r, s)|/(%1) + (n + 3 n q - 4 n s | 2 | \ds / 2 2 2 2 2 3 2 2 3 + 3 n q - 9 n q s + 6 n s + n q - 6 n q s + 9 n q s - 4 n s 3 2 2 3 4 3 2 2 2 - q s + 3 q s - 3 q s + s + 3 n + 7 n q - 9 n s + 5 n q 2 3 2 2 3 2 - 14 n q s + 9 n s + q - 5 q s + 7 q s - 3 s + 3 n + 5 n q - 6 n s / 3 \ 2 2 |d | + 2 q - 5 q s + 3 s + n + q - s) |--- A[n + 1](r, s)|/(%1) = 0 | 3 | \ds / 4 3 3 2 2 2 2 2 3 2 %1 := n + n q - 4 n s - 3 n q - 3 n q s + 6 n s - 5 n q + 6 n q s 2 3 4 3 2 2 3 4 3 2 + 3 n q s - 4 n s - 2 q + 5 q s - 3 q s - q s + s + 2 n + 6 n q 2 2 2 3 2 2 3 - 6 n s + 6 n q - 12 n q s + 6 n s + 2 q - 6 q s + 6 q s - 2 s 2 + 2 n q + 2 q - 2 q s - 2 n - q + 2 s - 1 3 2 2 2 3 2 3 2 %2 := n - 3 n s - 3 n q + 3 n s - 2 q + 3 q s - s + 3 n + 3 n q - 6 n s 2 - 3 q s + 3 s + 3 n + 2 q - 3 s + 1 and in Maple notation -(n^2*p^4-3*n^2*p^3*r+n^2*p^3*s+3*n^2*p^2*r^2-3*n^2*p^2*r*s-n^2*p*r^3+3*n^2*p*r ^2*s-n^2*r^3*s-2*n*p^5+6*n*p^4*r-2*n*p^4*s-6*n*p^3*r^2+6*n*p^3*r*s+2*n*p^2*r^3-\ 6*n*p^2*r^2*s+2*n*p*r^3*s+p^6-3*p^5*r+p^5*s+3*p^4*r^2-3*p^4*r*s-p^3*r^3+3*p^3*r ^2*s-p^2*r^3*s-8*n^2*p^3+17*n^2*p^2*r-7*n^2*p^2*s-10*n^2*p*r^2+14*n^2*p*r*s+n^2 *r^3-7*n^2*r^2*s+20*n*p^4-46*n*p^3*r+18*n*p^3*s+32*n*p^2*r^2-40*n*p^2*r*s-6*n*p *r^3+26*n*p*r^2*s-4*n*r^3*s-12*p^5+29*p^4*r-11*p^4*s-22*p^3*r^2+26*p^3*r*s+5*p^ 2*r^3-19*p^2*r^2*s+4*p*r^3*s+23*n^2*p^2-30*n^2*p*r+16*n^2*p*s+7*n^2*r^2-16*n^2* r*s-78*n*p^3+128*n*p^2*r-60*n*p^2*s-54*n*p*r^2+88*n*p*r*s+4*n*r^3-28*n*r^2*s+59 *p^4-110*p^3*r+48*p^3*s+59*p^2*r^2-84*p^2*r*s-8*p*r^3+40*p*r^2*s-4*r^3*s-28*n^2 *p+16*n^2*r-12*n^2*s+148*n*p^2-152*n*p*r+88*n*p*s+28*n*r^2-64*n*r*s-152*p^3+204 *p^2*r-104*p^2*s-68*p*r^2+120*p*r*s+4*r^3-28*r^2*s+12*n^2-136*n*p+64*n*r-48*n*s +216*p^2-184*p*r+112*p*s+28*r^2-64*r*s+48*n-160*p+64*r-48*s+48)/(n*p-n*r-n+p-r-\ 1)/(3*n*p^2-6*n*p*r+3*n*r^2+p^3-3*p*r^2+2*r^3-12*n*p+12*n*r-3*p^2-6*p*r+9*r^2+ 11*n-p+12*r+5)*A[n](r,s)+(2*n^2*p^4-5*n^2*p^3*r+3*n^2*p^3*s+3*n^2*p^2*r^2-9*n^2 *p^2*r*s+n^2*p*r^3+9*n^2*p*r^2*s-n^2*r^4-3*n^2*r^3*s-2*n*p^5+2*n*p^4*r-4*n*p^4* s+6*n*p^3*r^2+10*n*p^3*r*s-10*n*p^2*r^3-6*n*p^2*r^2*s+4*n*p*r^4-2*n*p*r^3*s+2*n *r^4*s+3*p^5*r+p^5*s-9*p^4*r^2-p^4*r*s+9*p^3*r^3-3*p^3*r^2*s-3*p^2*r^4+5*p^2*r^ 3*s-2*p*r^4*s-15*n^2*p^3+27*n^2*p^2*r-18*n^2*p^2*s-9*n^2*p*r^2+36*n^2*p*r*s-3*n ^2*r^3-18*n^2*r^2*s+22*n*p^4-22*n*p^3*r+36*n*p^3*s-30*n*p^2*r^2-72*n*p^2*r*s+38 *n*p*r^3+36*n*p*r^2*s-8*n*r^4-2*p^5-25*p^4*r-13*p^4*s+69*p^3*r^2+16*p^3*r*s-55* p^2*r^3+12*p^2*r^2*s+13*p*r^4-20*p*r^3*s+5*r^4*s+40*n^2*p^2-46*n^2*p*r+34*n^2*p *s+6*n^2*r^2-34*n^2*r*s-92*n*p^3+80*n*p^2*r-116*n*p^2*s+48*n*p*r^2+164*n*p*r*s-\ 36*n*r^3-48*n*r^2*s+20*p^4+76*p^3*r+64*p^3*s-192*p^2*r^2-76*p^2*r*s+110*p*r^3-6 *p*r^2*s-14*r^4+18*r^3*s-45*n^2*p+24*n^2*r-21*n^2*s+182*n*p^2-116*n*p*r+158*n*p *s-24*n*r^2-116*n*r*s-77*p^3-100*p^2*r-149*p^2*s+228*p*r^2+140*p*r*s-72*r^3-12* r^2*s+18*n^2-170*n*p+56*n*r-78*n*s+142*p^2+50*p*r+164*p*s-96*r^2-86*r*s+60*n-\ 125*p-4*r-69*s+42)/(n*p-n*r-n+p-r-1)/(3*n*p^2-6*n*p*r+3*n*r^2+p^3-3*p*r^2+2*r^3 -12*n*p+12*n*r-3*p^2-6*p*r+9*r^2+11*n-p+12*r+5)*diff(A[n](r,s),r)-(n^2*p^3+3*n^ 2*p^2*s-3*n^2*p*r^2-6*n^2*p*r*s+2*n^2*r^3+3*n^2*r^2*s-4*n*p^3*r-2*n*p^3*s+6*n*p ^2*r^2+6*n*p*r^2*s-2*n*r^4-4*n*r^3*s+3*p^3*r^2+2*p^3*r*s-6*p^2*r^3-3*p^2*r^2*s+ 3*p*r^4+r^4*s-6*n^2*p^2-12*n^2*p*s+6*n^2*r^2+12*n^2*r*s+2*n*p^3+24*n*p^2*r+18*n *p^2*s-30*n*p*r^2-12*n*p*r*s+4*n*r^3-6*n*r^2*s-4*p^3*r-2*p^3*s-12*p^2*r^2-12*p^ 2*r*s+24*p*r^3+18*p*r^2*s-8*r^4-4*r^3*s+11*n^2*p+11*n^2*s-12*n*p^2-44*n*p*r-46* n*p*s+34*n*r^2+24*n*r*s+p^3+24*p^2*r+15*p^2*s+6*p*r^2+16*p*r*s-20*r^3-20*r^2*s-\ 6*n^2+22*n*p+24*n*r+34*n*s-6*p^2-44*p*r-34*p*s+10*r^2-12*n+11*p+24*r+23*s-6)/(n +1)/(3*n*p^2-6*n*p*r+3*n*r^2+p^3-3*p*r^2+2*r^3-12*n*p+12*n*r-3*p^2-6*p*r+9*r^2+ 11*n-p+12*r+5)*diff(diff(A[n](r,s),r),r)+(n^2*p^2*r+n^2*p^2*s-2*n^2*p*r^2-2*n^2 *p*r*s+n^2*r^3+n^2*r^2*s-2*n*p^2*r^2-2*n*p^2*r*s+4*n*p*r^3+4*n*p*r^2*s-2*n*r^4-\ 2*n*r^3*s+p^2*r^3+p^2*r^2*s-2*p*r^4-2*p*r^3*s+r^5+r^4*s-3*n^2*p*r-3*n^2*p*s+3*n ^2*r^2+3*n^2*r*s+2*n*p^2*r+2*n*p^2*s+2*n*p*r^2+2*n*p*r*s-4*n*r^3-4*n*r^2*s-2*p^ 2*r^2-2*p^2*r*s+p*r^3+p*r^2*s+r^4+r^3*s+2*n^2*r+2*n^2*s-6*n*p*r-6*n*p*s+2*n*r^2 +2*n*r*s+p^2*r+p^2*s+4*p*r^2+4*p*r*s-3*r^3-3*r^2*s+4*n*r+4*n*s-3*p*r-3*p*s-r^2- r*s+2*r+2*s)/(n+1)/(3*n*p^2-6*n*p*r+3*n*r^2+p^3-3*p*r^2+2*r^3-12*n*p+12*n*r-3*p ^2-6*p*r+9*r^2+11*n-p+12*r+5)*diff(diff(diff(A[n](r,s),r),r),r)+(n*p^3-3*n*p^2* r+3*n*p*r^2-n*r^3+p^4-3*p^3*r+3*p^2*r^2-p*r^3-7*n*p^2+14*n*p*r-7*n*r^2-7*p^3+14 *p^2*r-7*p*r^2+16*n*p-16*n*r+16*p^2-16*p*r-12*n-12*p)/(3*n*p^2-6*n*p*r+3*n*r^2+ p^3-3*p*r^2+2*r^3-12*n*p+12*n*r-3*p^2-6*p*r+9*r^2+11*n-p+12*r+5)/(p-r-1)*A[n+1] (r,s)-(3*n*p^3-9*n*p^2*r+9*n*p*r^2-3*n*r^3+2*p^4-5*p^3*r+3*p^2*r^2+p*r^3-r^4-18 *n*p^2+36*n*p*r-18*n*r^2-12*p^3+18*p^2*r-6*r^3+34*n*p-34*n*r+22*p^2-10*p*r-12*r ^2-21*n-11*p-10*r-3)/(3*n*p^2-6*n*p*r+3*n*r^2+p^3-3*p*r^2+2*r^3-12*n*p+12*n*r-3 *p^2-6*p*r+9*r^2+11*n-p+12*r+5)/(p-r-1)*diff(A[n+1](r,s),r)+diff(diff(A[n+1](r, s),r),r)-(n*p^2-2*n*p*r+n*r^2+p^2*r-2*p*r^2+r^3-3*n*p+3*n*r+p^2-5*p*r+4*r^2+2*n -3*p+5*r+2)/(3*n*p^2-6*n*p*r+3*n*r^2+p^3-3*p*r^2+2*r^3-12*n*p+12*n*r-3*p^2-6*p* r+9*r^2+11*n-p+12*r+5)*diff(diff(diff(A[n+1](r,s),r),r),r) = 0 -(4*n^5+16*n^4*q-12*n^4*s+25*n^3*q^2-36*n^3*q*s+12*n^3*s^2+19*n^2*q^3-39*n^2*q^ 2*s+24*n^2*q*s^2-4*n^2*s^3+7*n*q^4-18*n*q^3*s+15*n*q^2*s^2-4*n*q*s^3+q^5-3*q^4* s+3*q^3*s^2-q^2*s^3+16*n^4+50*n^3*q-36*n^3*s+57*n^2*q^2-78*n^2*q*s+24*n^2*s^2+ 28*n*q^3-54*n*q^2*s+30*n*q*s^2-4*n*s^3+5*q^4-12*q^3*s+9*q^2*s^2-2*q*s^3+25*n^3+ 57*n^2*q-39*n^2*s+42*n*q^2-54*n*q*s+15*n*s^2+10*q^3-18*q^2*s+9*q*s^2-s^3+19*n^2 +28*n*q-18*n*s+10*q^2-12*q*s+3*s^2+7*n+5*q-3*s+1)/(n+1)/(n^3-3*n^2*s-3*n*q^2+3* n*s^2-2*q^3+3*q^2*s-s^3+3*n^2+3*n*q-6*n*s-3*q*s+3*s^2+3*n+2*q-3*s+1)*A[n](r,s)+ (4*n^6+18*n^5*q-12*n^5*s+32*n^4*q^2-38*n^4*q*s+8*n^4*s^2+28*n^3*q^3-40*n^3*q^2* s+4*n^3*q*s^2+8*n^3*s^3+12*n^2*q^4-12*n^2*q^3*s-24*n^2*q^2*s^2+36*n^2*q*s^3-12* n^2*s^4+2*n*q^5+4*n*q^4*s-28*n*q^3*s^2+40*n*q^2*s^3-22*n*q*s^4+4*n*s^5+2*q^5*s-\ 8*q^4*s^2+12*q^3*s^3-8*q^2*s^4+2*q*s^5+15*n^5+57*n^4*q-39*n^4*s+83*n^3*q^2-104* n^3*q*s+26*n^3*s^2+57*n^2*q^3-93*n^2*q^2*s+30*n^2*q*s^2+6*n^2*s^3+18*n*q^4-30*n *q^3*s-3*n*q^2*s^2+24*n*q*s^3-9*n*s^4+2*q^5-2*q^4*s-7*q^3*s^2+13*q^2*s^3-7*q*s^ 4+s^5+19*n^4+60*n^3*q-44*n^3*s+69*n^2*q^2-96*n^2*q*s+30*n^2*s^2+34*n*q^3-66*n*q ^2*s+36*n*q*s^2-4*n*s^3+6*q^4-14*q^3*s+9*q^2*s^2-s^4+6*n^3+18*n^2*q-18*n^2*s+17 *n*q^2-32*n*q*s+14*n*s^2+5*q^3-13*q^2*s+10*q*s^2-2*s^3-6*n^2-6*n*q-q^2-2*q*s+2* s^2-5*n-3*q+s-1)/(n+1)/(n^4+n^3*q-4*n^3*s-3*n^2*q^2-3*n^2*q*s+6*n^2*s^2-5*n*q^3 +6*n*q^2*s+3*n*q*s^2-4*n*s^3-2*q^4+5*q^3*s-3*q^2*s^2-q*s^3+s^4+2*n^3+6*n^2*q-6* n^2*s+6*n*q^2-12*n*q*s+6*n*s^2+2*q^3-6*q^2*s+6*q*s^2-2*s^3+2*n*q+2*q^2-2*q*s-2* n-q+2*s-1)*diff(A[n](r,s),s)-(n^6+4*n^5*q-2*n^5*s+6*n^4*q^2-4*n^4*q*s-n^4*s^2+4 *n^3*q^3-8*n^3*q*s^2+4*n^3*s^3+n^2*q^4+4*n^2*q^3*s-12*n^2*q^2*s^2+8*n^2*q*s^3-n ^2*s^4+2*n*q^4*s-4*n*q^3*s^2+4*n*q*s^4-2*n*s^5+q^4*s^2-4*q^3*s^3+6*q^2*s^4-4*q* s^5+s^6+5*n^5+17*n^4*q-9*n^4*s+21*n^3*q^2-16*n^3*q*s-2*n^3*s^2+11*n^2*q^3-3*n^2 *q^2*s-18*n^2*q*s^2+10*n^2*s^3+2*n*q^4+6*n*q^3*s-21*n*q^2*s^2+16*n*q*s^3-3*n*s^ 4+2*q^4*s-5*q^3*s^2+3*q^2*s^3+q*s^4-s^5+10*n^4+28*n^3*q-16*n^3*s+27*n^2*q^2-24* n^2*q*s+10*n*q^3-6*n*q^2*s-12*n*q*s^2+8*n*s^3+q^4+2*q^3*s-9*q^2*s^2+8*q*s^3-2*s ^4+10*n^3+22*n^2*q-14*n^2*s+15*n*q^2-16*n*q*s+2*n*s^2+3*q^3-3*q^2*s-2*q*s^2+2*s ^3+5*n^2+8*n*q-6*n*s+3*q^2-4*q*s+s^2+n+q-s)/(n^2+n*q-n*s+q-s-1)/(n^3-3*n^2*s-3* n*q^2+3*n*s^2-2*q^3+3*q^2*s-s^3+3*n^2+3*n*q-6*n*s-3*q*s+3*s^2+3*n+2*q-3*s+1)* diff(diff(A[n](r,s),s),s)+(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2)*q/(n^3-3*n^2*s-3*n*q^ 2+3*n*s^2-2*q^3+3*q^2*s-s^3+3*n^2+3*n*q-6*n*s-3*q*s+3*s^2+3*n+2*q-3*s+1)*A[n+1] (r,s)+diff(A[n+1](r,s),s)-(2*n^4+5*n^3*q-8*n^3*s+3*n^2*q^2-15*n^2*q*s+12*n^2*s^ 2-n*q^3-6*n*q^2*s+15*n*q*s^2-8*n*s^3-q^4+q^3*s+3*q^2*s^2-5*q*s^3+2*s^4+5*n^3+12 *n^2*q-15*n^2*s+9*n*q^2-24*n*q*s+15*n*s^2+2*q^3-9*q^2*s+12*q*s^2-5*s^3+3*n^2+7* n*q-6*n*s+4*q^2-7*q*s+3*s^2-n+s-1)/(n^4+n^3*q-4*n^3*s-3*n^2*q^2-3*n^2*q*s+6*n^2 *s^2-5*n*q^3+6*n*q^2*s+3*n*q*s^2-4*n*s^3-2*q^4+5*q^3*s-3*q^2*s^2-q*s^3+s^4+2*n^ 3+6*n^2*q-6*n^2*s+6*n*q^2-12*n*q*s+6*n*s^2+2*q^3-6*q^2*s+6*q*s^2-2*s^3+2*n*q+2* q^2-2*q*s-2*n-q+2*s-1)*diff(diff(A[n+1](r,s),s),s)+(n^4+3*n^3*q-4*n^3*s+3*n^2*q ^2-9*n^2*q*s+6*n^2*s^2+n*q^3-6*n*q^2*s+9*n*q*s^2-4*n*s^3-q^3*s+3*q^2*s^2-3*q*s^ 3+s^4+3*n^3+7*n^2*q-9*n^2*s+5*n*q^2-14*n*q*s+9*n*s^2+q^3-5*q^2*s+7*q*s^2-3*s^3+ 3*n^2+5*n*q-6*n*s+2*q^2-5*q*s+3*s^2+n+q-s)/(n^4+n^3*q-4*n^3*s-3*n^2*q^2-3*n^2*q *s+6*n^2*s^2-5*n*q^3+6*n*q^2*s+3*n*q*s^2-4*n*s^3-2*q^4+5*q^3*s-3*q^2*s^2-q*s^3+ s^4+2*n^3+6*n^2*q-6*n^2*s+6*n*q^2-12*n*q*s+6*n*s^2+2*q^3-6*q^2*s+6*q*s^2-2*s^3+ 2*n*q+2*q^2-2*q*s-2*n-q+2*s-1)*diff(diff(diff(A[n+1](r,s),s),s),s) = 0 ------------------------------------------------- This took, 0.257, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ A[n](r, s) = ) / ----- k = 0 3 (k - 1 + p) (n - k + q) k binomial(n, k) binomial(n + k, k) (r + k) (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)*binomial(n+k,k)^3*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x^k ,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 3 4 3 3 3 3 3 2 2 3 2 3 3 - (n p - 3 n p r + n p s + 3 n p r - 3 n p r s - n p r 3 2 3 3 2 5 2 4 2 4 2 3 2 + 3 n p r s - n r s - 3 n p + 9 n p r - 3 n p s - 9 n p r 2 3 2 2 3 2 2 2 2 3 6 + 9 n p r s + 3 n p r - 9 n p r s + 3 n p r s + 3 n p 5 5 4 2 4 3 3 3 2 - 9 n p r + 3 n p s + 9 n p r - 9 n p r s - 3 n p r + 9 n p r s 2 3 7 6 6 5 2 5 4 3 - 3 n p r s - p + 3 p r - p s - 3 p r + 3 p r s + p r 4 2 3 3 3 3 3 2 3 2 3 2 - 3 p r s + p r s - 10 n p + 21 n p r - 9 n p s - 12 n p r 3 3 3 3 2 2 4 2 3 2 3 + 18 n p r s + n r - 9 n r s + 36 n p - 81 n p r + 33 n p s 2 2 2 2 2 2 3 2 2 2 3 + 54 n p r - 72 n p r s - 9 n p r + 45 n p r s - 6 n r s 5 4 4 3 2 3 2 3 - 42 n p + 99 n p r - 39 n p s - 72 n p r + 90 n p r s + 15 n p r 2 2 3 6 5 5 4 2 - 63 n p r s + 12 n p r s + 16 p - 39 p r + 15 p s + 30 p r 4 3 3 3 2 2 3 3 2 3 - 36 p r s - 7 p r + 27 p r s - 6 p r s + 35 n p - 44 n p r 3 3 2 3 2 3 2 2 2 2 + 26 n p s + 9 n r - 26 n r s - 165 n p + 258 n p r - 132 n p s 2 2 2 2 3 2 2 4 3 - 99 n p r + 186 n p r s + 6 n r - 54 n r s + 237 n p - 420 n p r 3 2 2 2 3 2 + 198 n p s + 207 n p r - 330 n p r s - 24 n p r + 144 n p r s 3 5 4 4 3 2 3 - 12 n r s - 107 p + 206 p r - 92 p s - 117 p r + 170 p r s 2 3 2 2 3 3 3 3 + 18 p r - 90 p r s + 12 p r s - 50 n p + 26 n r - 24 n s 2 2 2 2 2 2 2 3 + 360 n p - 342 n p r + 228 n p s + 54 n r - 156 n r s - 690 n p 2 2 2 3 + 858 n p r - 492 n p s - 252 n p r + 528 n p r s + 12 n r 2 4 3 3 2 2 2 - 108 n r s + 388 p - 566 p r + 296 p s + 222 p r - 396 p r s 3 2 3 3 2 2 2 - 20 p r + 132 p r s - 8 r s + 24 n - 372 n p + 156 n r - 144 n s 2 2 3 + 1092 n p - 840 n p r + 600 n p s + 108 n r - 312 n r s - 824 p 2 2 2 3 2 2 + 852 p r - 528 p s - 204 p r + 456 p r s + 8 r - 72 r s + 144 n 2 2 - 888 n p + 312 n r - 288 n s + 1024 p - 664 p r + 496 p s + 72 r / - 208 r s + 288 n - 688 p + 208 r - 192 s + 192) A[n](r, s) / ( / 2 3 3 6 3 5 (n + 2 n + 1) (n + r + 1) (p - r - 1) ) + (3 n p - 14 n p r 3 5 3 4 2 3 4 3 3 3 3 3 2 + 4 n p s + 25 n p r - 20 n p r s - 20 n p r + 40 n p r s 3 2 4 3 2 3 3 5 3 4 3 6 + 5 n p r - 40 n p r s + 2 n p r + 20 n p r s - n r 3 5 2 7 2 6 2 6 2 5 2 - 4 n r s - 6 n p + 24 n p r - 9 n p s - 30 n p r 2 5 2 4 2 2 3 4 2 3 3 2 2 5 + 42 n p r s - 75 n p r s + 30 n p r + 60 n p r s - 24 n p r 2 2 4 2 6 2 5 2 6 8 7 - 15 n p r s + 6 n p r - 6 n p r s + 3 n r s + 3 n p - 6 n p r 7 6 2 6 5 3 5 2 + 6 n p s - 15 n p r - 24 n p r s + 60 n p r + 30 n p r s 4 4 3 5 3 4 2 6 2 5 - 75 n p r + 42 n p r - 30 n p r s - 9 n p r + 24 n p r s 6 8 8 7 2 7 6 3 6 2 - 6 n p r s - 4 p r - p s + 20 p r + 2 p r s - 40 p r + 5 p r s 5 4 5 3 4 5 4 4 3 6 3 5 + 40 p r - 20 p r s - 20 p r + 25 p r s + 4 p r - 14 p r s 2 6 3 5 3 4 3 4 3 3 2 + 3 p r s - 39 n p + 149 n p r - 46 n p s - 206 n p r 3 3 3 2 3 3 2 2 3 4 + 184 n p r s + 114 n p r - 276 n p r s - 11 n p r 3 3 3 5 3 4 2 6 2 5 + 184 n p r s - 7 n r - 46 n r s + 96 n p - 330 n p r 2 5 2 4 2 2 4 2 3 3 + 129 n p s + 348 n p r - 507 n p r s - 12 n p r 2 3 2 2 2 4 2 2 3 2 5 + 738 n p r s - 192 n p r - 462 n p r s + 102 n p r 2 4 2 6 2 5 7 6 6 + 93 n p r s - 12 n r + 9 n r s - 60 n p + 123 n p r - 105 n p s 5 2 5 4 3 4 2 3 4 + 147 n p r + 372 n p r s - 618 n p r - 423 n p r s + 642 n p r 3 3 2 5 2 4 6 5 + 72 n p r s - 273 n p r + 177 n p r s + 39 n p r - 108 n p r s 6 8 7 7 6 2 6 5 3 + 15 n r s + 3 p + 58 p r + 22 p s - 289 p r - 49 p r s + 516 p r 5 2 4 4 4 3 3 5 3 4 - 39 p r s - 439 p r + 206 p r s + 178 p r - 224 p r s 2 6 2 5 6 3 4 3 3 - 27 p r + 99 p r s - 15 p r s + 202 n p - 606 n p r 3 3 3 2 2 3 2 3 3 3 2 + 202 n p s + 606 n p r - 606 n p r s - 202 n p r + 606 n p r s 3 3 2 5 2 4 2 4 2 3 2 - 202 n r s - 636 n p + 1830 n p r - 744 n p s - 1566 n p r 2 3 2 2 3 2 2 2 2 4 + 2370 n p r s + 78 n p r - 2646 n p r s + 402 n p r 2 3 2 5 2 4 6 5 + 1158 n p r s - 108 n r - 138 n r s + 507 n p - 1005 n p r 5 4 2 4 3 3 3 2 + 765 n p s - 486 n p r - 2337 n p r s + 2460 n p r + 2304 n p r s 2 4 2 3 5 4 6 - 2019 n p r - 540 n p r s + 585 n p r - 309 n p r s - 42 n r 5 7 6 6 5 2 5 + 117 n r s - 54 p - 333 p r - 204 p s + 1731 p r + 459 p r s 4 3 4 2 3 4 3 3 2 5 - 2716 p r + 21 p r s + 1902 p r - 796 p r s - 591 p r 2 4 6 5 6 3 3 3 2 + 732 p r s + 61 p r - 231 p r s + 19 r s - 535 n p + 1178 n p r 3 2 3 2 3 3 3 3 2 - 427 n p s - 751 n p r + 854 n p r s + 108 n r - 427 n r s 2 4 2 3 2 3 2 2 2 + 2259 n p - 5220 n p r + 2211 n p s + 3387 n p r 2 2 2 3 2 2 2 4 2 3 - 5352 n p r s - 150 n p r + 4071 n p r s - 276 n r - 930 n r s 5 4 4 3 2 3 - 2361 n p + 4284 n p r - 3003 n p s + 447 n p r + 7590 n p r s 2 3 2 2 4 3 5 - 4716 n p r - 6033 n p r s + 2760 n p r + 1308 n p r s - 414 n r 4 6 5 5 4 2 4 + 138 n r s + 414 p + 919 p r + 1042 p s - 5543 p r - 2207 p r s 3 3 3 2 2 4 2 3 5 + 7448 p r + 619 p r s - 4059 p r + 1392 p r s + 867 p r 4 6 5 3 2 3 3 - 1023 p r s - 46 r + 177 r s + 767 n p - 1093 n p r + 441 n p s 3 2 3 2 3 2 2 2 2 + 326 n r - 441 n r s - 4644 n p + 8049 n p r - 3582 n p s 2 2 2 2 3 2 2 4 - 3489 n p r + 5841 n p r s + 84 n r - 2259 n r s + 6621 n p 3 3 2 2 2 3 - 10341 n p r + 6855 n p s + 762 n p r - 13401 n p r s + 4338 n p r 2 4 3 5 4 + 7560 n p r s - 1380 n r - 1014 n r s - 1764 p - 1009 p r 4 3 2 3 2 3 2 2 - 3208 p s + 10177 p r + 5977 p r s - 11186 p r - 2265 p r s 4 3 5 4 3 3 + 4256 p r - 1010 p r s - 474 r + 506 r s - 566 n p + 386 n r 3 2 2 2 2 2 2 - 180 n s + 5529 n p - 6339 n p r + 3021 n p s + 1350 n r 2 3 2 2 2 - 2481 n r s - 11445 n p + 14166 n p r - 9111 n p s - 1737 n p r 3 2 4 3 3 + 12180 n p r s - 1524 n r - 3609 n r s + 4564 p - 711 p r + 6100 p s 2 2 2 3 2 4 3 - 10611 p r - 9189 p r s + 8686 p r + 3099 p r s - 1748 r + 170 r s 3 2 2 2 2 + 168 n - 3534 n p + 1986 n r - 1044 n s + 11913 n p - 10203 n p r 2 3 2 2 + 6555 n p s + 882 n r - 4467 n r s - 7336 p + 3087 p r - 7008 p s 2 3 2 2 + 5745 p r + 7461 p r s - 2708 r - 1497 r s + 936 n - 6834 n p 2 2 + 2982 n r - 1980 n s + 7151 p - 3005 p r + 4463 p s - 1230 r - 2483 r s /d \ / + 1656 n - 3866 p + 998 r - 1212 s + 888) |-- A[n](r, s)| / ( \dr / / 2 2 2 (n p - n r - n + 2 n p - 2 n r - 2 n + p - r - 1) 2 2 2 2 2 2 (p - 2 p r + r - 3 p + 3 r + 2) (n p - 2 n p r + n r + p r - 2 p r 3 2 2 + r - 3 n p + 3 n r + p - 5 p r + 4 r + 2 n - 3 p + 5 r + 2)) - ( 3 6 3 5 3 5 3 4 2 3 4 3 n p - 12 n p r + 6 n p s + 15 n p r - 30 n p r s 3 3 2 3 2 4 3 2 3 3 5 3 4 + 60 n p r s - 15 n p r - 60 n p r s + 12 n p r + 30 n p r s 3 6 3 5 2 7 2 6 2 6 2 5 2 - 3 n r - 6 n r s - 3 n p + 3 n p r - 9 n p s + 27 n p r 2 5 2 4 3 2 4 2 2 3 4 2 2 5 + 36 n p r s - 75 n p r - 45 n p r s + 75 n p r - 27 n p r 2 2 4 2 6 2 5 2 7 2 6 + 45 n p r s - 3 n p r - 36 n p r s + 3 n r + 9 n r s 7 7 6 2 6 5 3 + 9 n p r + 3 n p s - 36 n p r - 3 n p r s + 45 n p r 5 2 4 3 3 5 3 4 2 6 - 27 n p r s + 75 n p r s - 45 n p r - 75 n p r s + 36 n p r 2 5 7 6 7 7 2 7 + 27 n p r s - 9 n p r + 3 n p r s - 3 n r s - 6 p r - 3 p r s 6 3 6 2 5 4 5 3 4 5 3 6 + 30 p r + 12 p r s - 60 p r - 15 p r s + 60 p r - 30 p r 3 5 2 7 2 6 7 3 5 3 4 + 15 p r s + 6 p r - 12 p r s + 3 p r s - 42 n p + 138 n p r 3 4 3 3 2 3 3 3 2 3 3 2 2 - 72 n p s - 132 n p r + 288 n p r s - 12 n p r - 432 n p r s 3 4 3 3 3 5 3 4 2 6 + 78 n p r + 288 n p r s - 30 n r - 72 n r s + 57 n p 2 5 2 5 2 4 2 2 4 2 3 3 - 72 n p r + 144 n p s - 279 n p r - 504 n p r s + 696 n p r 2 3 2 2 2 4 2 2 3 2 5 + 576 n p r s - 549 n p r - 144 n p r s + 144 n p r 2 4 2 6 2 5 7 6 6 - 144 n p r s + 3 n r + 72 n r s - 6 n p - 138 n p r - 66 n p s 5 2 5 4 3 4 2 3 4 + 540 n p r + 108 n p r s - 636 n p r + 234 n p r s + 114 n p r 3 3 2 5 2 4 6 5 - 696 n p r s + 270 n p r + 594 n p r s - 168 n p r - 180 n p r s 7 6 7 7 6 2 6 5 3 + 24 n r + 6 n r s + 9 p r + 3 p s + 60 p r + 45 p r s - 363 p r 5 2 4 4 4 3 3 5 3 4 - 189 p r s + 672 p r + 237 p r s - 573 p r - 63 p r s 2 6 2 5 7 6 7 3 4 + 228 p r - 81 p r s - 33 p r + 57 p r s - 9 r s + 236 n p 3 3 3 3 3 2 2 3 2 3 3 - 611 n p r + 333 n p s + 417 n p r - 999 n p r s + 55 n p r 3 2 3 4 3 3 2 5 2 4 + 999 n p r s - 97 n r - 333 n r s - 444 n p + 588 n p r 2 4 2 3 2 2 3 2 2 3 - 924 n p s + 1089 n p r + 2697 n p r s - 2355 n p r 2 2 2 2 4 2 3 2 5 2 4 - 2547 n p r s + 1311 n p r + 699 n p r s - 189 n r + 75 n r s 6 5 5 4 2 4 + 105 n p + 846 n p r + 588 n p s - 3249 n p r - 1092 n p r s 3 3 3 2 2 4 2 3 5 + 3435 n p r - 513 n p r s - 846 n p r + 2211 n p r s - 477 n p r 4 6 5 7 6 6 - 1455 n p r s + 186 n r + 261 n r s - 3 p - 141 p r - 57 p s 5 2 5 4 3 4 2 3 4 - 123 p r - 246 p r s + 1675 p r + 1161 p r s - 2878 p r 3 3 2 5 2 4 6 5 7 - 1377 p r s + 1992 p r + 480 p r s - 568 p r + 99 p r s + 46 r 6 3 3 3 2 3 2 3 2 - 60 r s - 680 n p + 1296 n p r - 744 n p s - 552 n p r 3 3 3 3 2 2 4 2 3 + 1488 n p r s - 64 n r - 744 n r s + 1836 n p - 2265 n p r 2 3 2 2 2 2 2 2 3 + 3039 n p s - 1989 n p r - 6885 n p r s + 3429 n p r 2 2 2 4 2 3 5 4 + 4653 n p r s - 1011 n r - 807 n r s - 762 n p - 2622 n p r 4 3 2 3 2 3 - 2760 n p s + 9990 n p r + 4962 n p r s - 8850 n p r 2 2 4 3 5 4 - 558 n p r s + 2028 n p r - 2730 n p r s + 216 n r + 1086 n r s 6 5 5 4 2 4 3 3 + 51 p + 906 p r + 450 p s - 699 p r + 510 p r s - 3565 p r 3 2 2 4 2 3 5 4 - 3501 p r s + 5808 p r + 3687 p r s - 2961 p r - 1161 p r s 6 5 3 2 3 3 3 2 + 460 r + 15 r s + 1057 n p - 1309 n p r + 805 n p s + 252 n r 3 2 3 2 2 2 2 2 2 - 805 n r s - 4347 n p + 4467 n p r - 5403 n p s + 1692 n p r 2 2 3 2 2 4 3 + 8391 n p r s - 1812 n r - 2988 n r s + 2964 n p + 4224 n p r 3 2 2 2 3 + 7386 n p s - 16479 n p r - 11352 n p r s + 10845 n p r 2 4 3 5 4 4 + 2961 n p r s - 1554 n r + 1005 n r s - 360 p - 3072 p r - 1908 p s 3 2 3 2 3 2 2 4 + 4155 p r + 246 p r s + 3107 p r + 5307 p r s - 5396 p r 3 5 4 3 3 3 - 4525 p r s + 1566 r + 880 r s - 838 n p + 498 n r - 340 n s 2 2 2 2 2 2 2 + 5883 n p - 4323 n p r + 4929 n p s - 540 n r - 3909 n r s 3 2 2 2 - 6654 n p - 3090 n p r - 11286 n p s + 13770 n p r + 12714 n p r s 3 2 4 3 3 2 2 - 5046 n r - 2448 n r s + 1364 p + 5878 p r + 4680 p s - 8649 p r 2 3 2 4 3 3 - 2754 p r s - 25 p r - 3603 p r s + 1772 r + 2017 r s + 264 n 2 2 2 2 - 4206 n p + 1602 n r - 1812 n s + 8595 n p + 357 n p r + 9135 n p s 2 3 2 2 2 - 4536 n r - 5511 n r s - 2987 p - 6261 p r - 6627 p s + 8142 p r 3 2 2 + 4119 p r s - 970 r + 696 r s + 1224 n - 5898 n p + 414 n r - 3036 n s 2 2 + 3769 p + 3371 p r + 5011 p s - 2880 r - 1975 r s + 1656 n - 2530 p / 2 \ |d | / 2 2 2 2 2 - 690 r - 1564 s + 696) |--- A[n](r, s)| / ((n p - 2 n p r + n r | 2 | / \dr / 2 2 2 2 2 2 - 3 n p + 3 n r + 2 n p - 4 n p r + 2 n r + 2 n - 6 n p + 6 n r + p 2 3 4 3 3 3 2 2 - 2 p r + r + 4 n - 3 p + 3 r + 2) %1) + (n p + 4 n p s - 6 n p r 3 2 3 3 3 2 3 4 3 3 - 12 n p r s + 8 n p r + 12 n p r s - 3 n r - 4 n r s 2 4 2 4 2 3 2 2 2 2 2 4 - 6 n p r - 3 n p s + 12 n p r + 18 n p r s - 12 n p r 2 3 2 5 2 4 4 2 4 3 3 - 24 n p r s + 6 n r + 9 n r s + 9 n p r + 6 n p r s - 24 n p r 3 2 2 4 4 6 5 4 3 - 12 n p r s + 18 n p r + 12 n p r s - 3 n r - 6 n r s - 4 p r 4 2 3 4 3 3 2 5 2 4 6 6 - 3 p r s + 12 p r + 8 p r s - 12 p r - 6 p r s + 4 p r + r s 3 3 3 2 3 2 3 3 3 3 2 - 10 n p - 30 n p s + 30 n p r + 60 n p r s - 20 n r - 30 n r s 2 4 2 3 2 3 2 2 2 2 2 + 3 n p + 60 n p r + 42 n p s - 108 n p r - 36 n p r s 2 3 2 2 2 4 2 3 4 4 + 24 n p r - 54 n p r s + 21 n r + 48 n r s - 12 n p r - 6 n p s 3 2 3 2 3 2 2 4 - 66 n p r - 60 n p r s + 180 n p r + 126 n p r s - 114 n p r 3 5 4 4 2 4 3 3 - 48 n p r s + 12 n r - 12 n r s + 9 p r + 6 p r s + 16 p r 3 2 2 4 2 3 5 4 6 + 18 p r s - 72 p r - 60 p r s + 60 p r + 42 p r s - 13 r 5 3 2 3 3 2 3 2 3 - 6 r s + 35 n p + 70 n p s - 35 n r - 70 n r s - 30 n p 2 2 2 2 2 2 2 2 3 - 210 n p r - 195 n p s + 300 n p r + 180 n p r s - 60 n r 2 2 4 3 3 2 2 + 15 n r s + 3 n p + 120 n p r + 72 n p s + 117 n p r 2 3 2 4 3 + 174 n p r s - 396 n p r - 354 n p r s + 156 n r + 108 n r s 4 4 3 2 3 2 3 2 2 4 - 6 p r - 3 p s - 78 p r - 60 p r s + 40 p r + 3 p r s + 108 p r 3 5 4 3 3 2 2 + 116 p r s - 64 r - 56 r s - 50 n p - 50 n s + 105 n p 2 2 2 2 2 3 2 + 300 n p r + 360 n p s - 255 n r - 210 n r s - 30 n p - 420 n p r 2 2 3 2 4 - 300 n p s + 60 n p r - 120 n p r s + 240 n r + 270 n r s + p 3 3 2 2 2 3 2 + 60 p r + 34 p s + 219 p r + 198 p r s - 212 p r - 138 p r s 4 3 3 2 2 2 2 - 18 r - 44 r s + 24 n - 150 n p - 144 n r - 222 n s + 105 n p 2 3 2 + 600 n p r + 510 n p s - 189 n r - 66 n r s - 10 p - 210 p r 2 2 3 2 2 - 135 p s - 210 p r - 240 p r s + 184 r + 153 r s + 72 n - 150 n p 2 2 - 288 n r - 294 n s + 35 p + 300 p r + 220 p s + 31 r + 74 r s + 72 n / 3 \ |d | / 2 3 2 2 - 50 p - 144 r - 122 s + 24) |--- A[n](r, s)| / ((n p - 3 n p r | 3 | / \dr / 2 2 2 3 2 2 2 2 2 3 2 + 3 n p r - n r - 6 n p + 12 n p r - 6 n r + 2 n p - 6 n p r 2 3 2 2 2 2 + 6 n p r - 2 n r + 11 n p - 11 n r - 12 n p + 24 n p r - 12 n r 3 2 2 3 2 2 + p - 3 p r + 3 p r - r - 6 n + 22 n p - 22 n r - 6 p + 12 p r 2 3 3 2 2 - 6 r - 12 n + 11 p - 11 r - 6) (n + r + 1)) - (n r + n s - 3 n r 2 3 2 4 3 2 2 2 - 3 n r s + 3 n r + 3 n r s - r - r s + 3 n r + 3 n s - 6 n r 3 2 2 - 6 n r s + 3 r + 3 r s + 3 n r + 3 n s - 3 r - 3 r s + r + s) / 4 \ |d | / 2 3 2 |--- A[n](r, s)| / ((n + 2 n + 1) (n + r + 1)) + (n p - 3 n p r | 4 | / \dr / 2 3 4 3 2 2 3 2 + 3 n p r - n r + p - 3 p r + 3 p r - p r - 9 n p + 18 n p r 2 3 2 2 2 - 9 n r - 9 p + 18 p r - 9 p r + 26 n p - 26 n r + 26 p - 26 p r / - 24 n - 24 p) A[n + 1](r, s) / ( / 2 2 5 (n p - n r + p r - r - n + p - 2 r - 1) (p - r - 1) ) - (4 n p 4 3 2 2 3 4 5 6 - 20 n p r + 40 n p r - 40 n p r + 20 n p r - 4 n r + 3 p 5 4 2 3 3 2 4 5 6 4 - 14 p r + 25 p r - 20 p r + 5 p r + 2 p r - r - 46 n p 3 2 2 3 4 5 4 + 184 n p r - 276 n p r + 184 n p r - 46 n r - 35 p + 129 p r 3 2 2 3 4 5 3 2 - 166 p r + 74 p r + 9 p r - 11 r + 202 n p - 606 n p r 2 3 4 3 2 2 3 4 + 606 n p r - 202 n r + 156 p - 422 p r + 330 p r - 18 p r - 46 r 2 2 3 2 2 3 - 427 n p + 854 n p r - 427 n r - 333 p + 572 p r - 145 p r - 94 r 2 2 + 441 n p - 441 n r + 340 p - 239 p r - 101 r - 180 n - 125 p - 55 r /d \ / - 12) |-- A[n + 1](r, s)| / ((p - r - 1) \dr / / 2 2 2 2 2 2 (p - 2 p r + r - 3 p + 3 r + 2) (n p - 2 n p r + n r + p r - 2 p r 3 2 2 5 + r - 3 n p + 3 n r + p - 5 p r + 4 r + 2 n - 3 p + 5 r + 2)) + (6 n p 4 3 2 2 3 4 5 6 - 30 n p r + 60 n p r - 60 n p r + 30 n p r - 6 n r + 3 p 5 4 2 2 4 5 6 4 3 - 12 p r + 15 p r - 15 p r + 12 p r - 3 r - 72 n p + 288 n p r 2 2 3 4 5 4 3 2 - 432 n p r + 288 n p r - 72 n r - 36 p + 108 p r - 72 p r 2 3 4 5 3 2 2 - 72 p r + 108 p r - 36 r + 333 n p - 999 n p r + 999 n p r 3 4 3 2 2 3 4 2 - 333 n r + 164 p - 323 p r - 15 p r + 343 p r - 169 r - 744 n p 2 3 2 2 3 + 1488 n p r - 744 n r - 347 p + 297 p r + 447 p r - 397 r + 805 n p 2 2 - 805 n r + 313 p + 179 p r - 492 r - 340 n - 33 p - 307 r - 76) / 2 \ |d | / 2 2 3 |--- A[n + 1](r, s)| / (%1 (p - 2 p r + r - 3 p + 3 r + 2)) - (4 n p | 2 | / \dr / 2 2 3 4 2 2 3 4 2 - 12 n p r + 12 n p r - 4 n r + p - 6 p r + 8 p r - 3 r - 30 n p 2 3 2 2 3 + 60 n p r - 30 n r - 6 p - 12 p r + 42 p r - 24 r + 70 n p - 70 n r / 3 \ 2 2 |d | + 5 p + 60 p r - 65 r - 50 n + 20 p - 70 r - 26) |--- A[n + 1](r, s)|/( | 3 | \dr / / 4 \ |d | %1) + |--- A[n + 1](r, s)| = 0 | 4 | \dr / 3 2 2 3 3 2 2 3 4 2 %1 := n p - 3 n p r + 3 n p r - n r + p r - 3 p r + 3 p r - r - 6 n p 2 3 2 2 3 + 12 n p r - 6 n r + p - 9 p r + 15 p r - 7 r + 11 n p - 11 n r 2 2 - 6 p + 23 p r - 17 r - 6 n + 11 p - 17 r - 6 6 5 5 4 2 4 4 2 3 3 (8 n + 36 n q - 24 n s + 66 n q - 84 n q s + 24 n s + 63 n q 3 2 3 2 3 3 2 4 2 3 2 2 2 - 114 n q s + 60 n q s - 8 n s + 33 n q - 75 n q s + 54 n q s 2 3 5 4 3 2 2 3 6 5 - 12 n q s + 9 n q - 24 n q s + 21 n q s - 6 n q s + q - 3 q s 4 2 3 3 5 4 4 3 2 3 3 2 + 3 q s - q s - 4 n - 8 n q - 4 n s - n q - 28 n q s + 20 n s 2 3 2 2 2 2 2 3 4 3 + 7 n q - 45 n q s + 48 n q s - 12 n s + 5 n q - 26 n q s 2 2 3 5 4 3 2 2 3 4 + 33 n q s - 12 n q s + q - 5 q s + 7 q s - 3 q s - 26 n 3 3 2 2 2 2 2 3 2 - 71 n q + 38 n s - 69 n q + 63 n q s - 6 n s - 28 n q + 30 n q s 2 3 4 3 2 2 3 3 2 + 3 n q s - 6 n s - 4 q + 4 q s + 3 q s - 3 q s - 7 n - 27 n q 2 2 2 3 2 2 3 + 33 n s - 24 n q + 42 n q s - 9 n s - 6 q + 12 q s - 3 q s - s 2 2 2 + 16 n + 11 n q + 10 n s + q + 7 q s - 2 s + 11 n + 5 q + s + 2) / 2 3 2 2 2 A[n](r, s) / ((n + 2 n + 1) (n + 2 n q - 3 n s + n q - 4 n q s / 2 2 2 3 2 2 2 + 3 n s - q s + 2 q s - s + n + 2 n q - 2 n s + q - 2 q s + s )) - ( 7 6 6 5 2 5 5 2 4 3 12 n + 60 n q - 36 n s + 123 n q - 132 n q s + 24 n s + 132 n q 4 2 4 2 4 3 3 4 3 3 - 177 n q s + 24 n q s + 24 n s + 78 n q - 96 n q s 3 2 2 3 3 3 4 2 5 2 4 - 66 n q s + 120 n q s - 36 n s + 24 n q - 6 n q s 2 3 2 2 2 3 2 4 2 5 6 5 - 120 n q s + 174 n q s - 84 n q s + 12 n s + 3 n q + 12 n q s 4 2 3 3 2 4 5 6 5 2 - 66 n q s + 96 n q s - 57 n q s + 12 n q s + 3 q s - 12 q s 4 3 3 4 2 5 6 5 5 4 2 + 18 q s - 12 q s + 3 q s - 6 n - 9 n q - 18 n s + 21 n q 4 4 2 3 3 3 2 3 2 3 3 - 129 n q s + 84 n s + 60 n q - 276 n q s + 294 n q s - 84 n s 2 4 2 3 2 2 2 2 3 2 4 5 + 54 n q - 252 n q s + 342 n q s - 162 n q s + 18 n s + 21 n q 4 3 2 2 3 4 5 6 - 102 n q s + 156 n q s - 84 n q s + 3 n q s + 6 n s + 3 q 5 4 2 3 3 2 4 5 5 4 - 15 q s + 24 q s - 12 q s - 3 q s + 3 q s - 41 n - 145 n q 4 3 2 3 3 2 2 3 2 2 + 85 n s - 194 n q + 196 n q s - 26 n s - 120 n q + 138 n q s 2 2 2 3 4 3 2 2 3 + 18 n q s - 38 n s - 33 n q + 24 n q s + 66 n q s - 76 n q s 4 5 4 3 2 2 3 4 5 4 + 19 n s - 3 q - 3 q s + 24 q s - 26 q s + 7 q s + s - 4 n 3 3 2 2 2 2 2 3 - 40 n q + 64 n s - 72 n q + 168 n q s - 72 n s - 45 n q 2 2 3 4 3 2 2 3 + 126 n q s - 84 n q s + 8 n s - 9 q + 27 q s - 18 q s - 4 q s 4 3 2 2 2 2 3 2 + 4 s + 28 n + 54 n q - 24 n s + 27 n q - 24 n s + 3 q + 9 q s 2 3 2 2 2 - 18 q s + 4 s + 4 n + 17 n q - 26 n s + 7 q - 11 q s - 2 s - 7 n - q /d \ / - 5 s - 2) |-- A[n](r, s)| / ( \ds / / 3 2 2 2 (n + n q - n s + 2 n + 2 n q - 2 n s + n + q - s) 2 2 6 (n + n q - 2 n s - q s + s + n + q - s) (-s - 1 + n + q)) + (6 n 5 5 4 2 4 4 2 3 3 + 21 n q - 6 n s + 27 n q - 3 n q s - 12 n s + 15 n q 3 2 3 2 3 3 2 4 2 3 2 2 2 + 18 n q s - 42 n q s + 12 n s + 3 n q + 21 n q s - 36 n q s 2 3 2 4 4 3 2 2 3 4 + 6 n q s + 6 n s + 6 n q s - 3 n q s - 18 n q s + 21 n q s 5 4 2 3 3 2 4 5 5 4 4 - 6 n s + 3 q s - 9 q s + 9 q s - 3 q s + 6 n + 27 n q - 24 n s 3 2 3 3 2 2 3 2 2 2 2 + 42 n q - 60 n q s + 12 n s + 27 n q - 36 n q s - 18 n q s 2 3 4 3 2 2 3 4 4 + 24 n s + 6 n q + 6 n q s - 54 n q s + 60 n q s - 18 n s + 6 q s 3 2 2 3 4 4 3 3 2 2 - 21 q s + 24 q s - 9 q s - 16 n - 25 n q - 14 n s - 3 n q 2 2 2 3 2 2 3 4 - 63 n q s + 42 n s + 9 n q - 60 n q s + 57 n q s - 10 n s + 3 q 3 2 2 3 4 3 2 2 2 - 15 q s + 15 q s - q s - 2 s - 22 n - 45 n q + 24 n s - 24 n q 2 3 2 2 3 + 6 n q s + 18 n s - 3 q - 6 q s + 15 q s - 4 s - 12 n q + 24 n s / 2 \ 2 |d | / 4 3 - 6 q + 12 q s + 8 n + 2 q + 4 s + 2) |--- A[n](r, s)| / ((n + 2 n q | 2 | / \ds / 3 2 2 2 2 2 3 2 2 2 - 2 n s + n q - 2 n q s + n s + n + 3 n q - 3 n s + 2 n q 2 2 2 2 4 - 4 n q s + 2 n s - n + q - 2 q s + s - n - q + s) (n - s + 1)) - (n 3 3 2 2 3 3 4 3 2 + n q + 2 n s + 3 n q s + 3 n q s - 2 n s + q s - s + 4 n + 3 n q 2 2 3 2 + 6 n s + 6 n q s + 3 q s - 2 s + 6 n + 3 n q + 6 n s + 3 q s + 4 n / 3 \ |d | / 2 3 + q + 2 s + 1) |--- A[n](r, s)| / ((n + 2 n + 1) (n - s + 1)) - q (n | 3 | / \ds / 2 2 2 2 3 2 2 3 + 3 n q - 3 n s + 3 n q - 6 n q s + 3 n s + q - 3 q s + 3 q s - s 2 2 2 - 3 n - 6 n q + 6 n s - 3 q + 6 q s - 3 s + 2 n + 2 q - 2 s) / 2 2 A[n + 1](r, s) / ((n + n q - 2 n s - q s + s + 2 n + q - 2 s + 1) / 2 2 2 6 (n + 2 n q - 2 n s + q - 2 q s + s + 2 n + 2 q - 2 s + 1)) - (n 5 5 4 2 4 4 2 3 3 3 2 + 2 n q - 6 n s - 5 n q - 10 n q s + 15 n s - 20 n q + 20 n q s 3 2 3 3 2 4 2 3 2 2 2 2 3 + 20 n q s - 20 n s - 25 n q + 60 n q s - 30 n q s - 20 n q s 2 4 5 4 3 2 2 3 4 + 15 n s - 14 n q + 50 n q s - 60 n q s + 20 n q s + 10 n q s 5 6 5 4 2 3 3 2 4 5 6 - 6 n s - 3 q + 14 q s - 25 q s + 20 q s - 5 q s - 2 q s + s 5 4 4 3 2 3 3 2 2 3 + n + 11 n q - 5 n s + 34 n q - 44 n q s + 10 n s + 46 n q 2 2 2 2 2 3 4 3 2 2 - 102 n q s + 66 n q s - 10 n s + 29 n q - 92 n q s + 102 n q s 3 4 5 4 3 2 2 3 4 5 - 44 n q s + 5 n s + 7 q - 29 q s + 46 q s - 34 q s + 11 q s - s 4 3 3 2 2 2 2 2 3 - 4 n - 10 n q + 16 n s - 6 n q + 30 n q s - 24 n s + 2 n q 2 2 3 4 3 2 2 3 + 12 n q s - 30 n q s + 16 n s + 2 q - 2 q s - 6 q s + 10 q s 4 3 2 2 2 2 3 - 4 s - 6 n - 19 n q + 18 n s - 20 n q + 38 n q s - 18 n s - 7 q 2 2 3 2 2 2 + 20 q s - 19 q s + 6 s + n - 3 n q - 2 n s - 4 q + 3 q s + s + 5 n /d \ / 6 5 5 4 2 + 3 q - 5 s + 2) |-- A[n + 1](r, s)| / (n + 5 n q - 6 n s + 10 n q \ds / / 4 4 2 3 3 3 2 3 2 3 3 - 25 n q s + 15 n s + 10 n q - 40 n q s + 50 n q s - 20 n s 2 4 2 3 2 2 2 2 3 2 4 5 + 5 n q - 30 n q s + 60 n q s - 50 n q s + 15 n s + n q 4 3 2 2 3 4 5 5 - 10 n q s + 30 n q s - 40 n q s + 25 n q s - 6 n s - q s 4 2 3 3 2 4 5 6 5 4 4 + 5 q s - 10 q s + 10 q s - 5 q s + s + 4 n + 17 n q - 20 n s 3 2 3 3 2 2 3 2 2 2 2 + 28 n q - 68 n q s + 40 n s + 22 n q - 84 n q s + 102 n q s 2 3 4 3 2 2 3 4 5 - 40 n s + 8 n q - 44 n q s + 84 n q s - 68 n q s + 20 n s + q 4 3 2 2 3 4 5 4 3 3 - 8 q s + 22 q s - 28 q s + 17 q s - 4 s + 6 n + 21 n q - 24 n s 2 2 2 2 2 3 2 2 + 27 n q - 63 n q s + 36 n s + 15 n q - 54 n q s + 63 n q s 3 4 3 2 2 3 4 3 2 - 24 n s + 3 q - 15 q s + 27 q s - 21 q s + 6 s + 4 n + 11 n q 2 2 2 3 2 2 3 - 12 n s + 10 n q - 22 n q s + 12 n s + 3 q - 10 q s + 11 q s - 4 s 2 2 2 6 5 5 + n + 2 n q - 2 n s + q - 2 q s + s ) + (3 n + 12 n q - 18 n s 4 2 4 4 2 3 2 3 2 3 3 + 15 n q - 60 n q s + 45 n s - 60 n q s + 120 n q s - 60 n s 2 4 2 2 2 2 3 2 4 5 4 - 15 n q + 90 n q s - 120 n q s + 45 n s - 12 n q + 30 n q s 2 3 4 5 6 5 4 2 2 4 - 60 n q s + 60 n q s - 18 n s - 3 q + 12 q s - 15 q s + 15 q s 5 6 4 3 2 3 2 3 2 2 - 12 q s + 3 s + 12 n q + 48 n q - 48 n q s + 72 n q - 144 n q s 2 2 4 3 2 2 3 5 + 72 n q s + 48 n q - 144 n q s + 144 n q s - 48 n q s + 12 q 4 3 2 2 3 4 4 3 3 - 48 q s + 72 q s - 48 q s + 12 q s - 11 n - 41 n q + 44 n s 2 2 2 2 2 3 2 2 - 57 n q + 123 n q s - 66 n s - 35 n q + 114 n q s - 123 n q s 3 4 3 2 2 3 4 3 2 + 44 n s - 8 q + 35 q s - 57 q s + 41 q s - 11 s - 5 n - 21 n q 2 2 2 3 2 2 + 15 n s - 27 n q + 42 n q s - 15 n s - 11 q + 27 q s - 21 q s 3 2 2 2 + 5 s + 6 n + 11 n q - 12 n s + 5 q - 11 q s + 6 s + n + 3 q - s - 2) / 2 \ |d | / 5 4 4 3 2 |--- A[n + 1](r, s)| / ((-s - 1 + n + q) (n + 4 n q - 5 n s + 6 n q | 2 | / \ds / 3 3 2 2 3 2 2 2 2 2 3 - 16 n q s + 10 n s + 4 n q - 18 n q s + 24 n q s - 10 n s 4 3 2 2 3 4 4 3 2 + n q - 8 n q s + 18 n q s - 16 n q s + 5 n s - q s + 4 q s 2 3 4 5 4 3 3 2 2 2 - 6 q s + 4 q s - s + 3 n + 10 n q - 12 n s + 12 n q - 30 n q s 2 2 3 2 2 3 4 3 + 18 n s + 6 n q - 24 n q s + 30 n q s - 12 n s + q - 6 q s 2 2 3 4 3 2 2 2 + 12 q s - 10 q s + 3 s + 3 n + 8 n q - 9 n s + 7 n q - 16 n q s 2 3 2 2 3 2 2 + 9 n s + 2 q - 7 q s + 8 q s - 3 s + n + 2 n q - 2 n s + q - 2 q s 2 4 3 3 2 2 2 2 2 + s )) - (3 n + 8 n q - 12 n s + 6 n q - 24 n q s + 18 n s 2 2 3 4 2 2 3 4 2 - 12 n q s + 24 n q s - 12 n s - q + 6 q s - 8 q s + 3 s + 6 n q 2 3 2 2 2 + 12 n q - 12 n q s + 6 q - 12 q s + 6 q s - 7 n - 12 n q + 14 n s / 3 \ 2 2 |d | / - 5 q + 12 q s - 7 s - 2 n - 4 q + 2 s + 2) |--- A[n + 1](r, s)| / ( | 3 | / \ds / 2 2 (n + q - s) (n + n q - 2 n s - q s + s + 2 n + q - 2 s + 1) / 4 \ |d | (-s - 1 + n + q)) + |--- A[n + 1](r, s)| = 0 | 4 | \ds / and in Maple notation -(n^3*p^4-3*n^3*p^3*r+n^3*p^3*s+3*n^3*p^2*r^2-3*n^3*p^2*r*s-n^3*p*r^3+3*n^3*p*r ^2*s-n^3*r^3*s-3*n^2*p^5+9*n^2*p^4*r-3*n^2*p^4*s-9*n^2*p^3*r^2+9*n^2*p^3*r*s+3* n^2*p^2*r^3-9*n^2*p^2*r^2*s+3*n^2*p*r^3*s+3*n*p^6-9*n*p^5*r+3*n*p^5*s+9*n*p^4*r ^2-9*n*p^4*r*s-3*n*p^3*r^3+9*n*p^3*r^2*s-3*n*p^2*r^3*s-p^7+3*p^6*r-p^6*s-3*p^5* r^2+3*p^5*r*s+p^4*r^3-3*p^4*r^2*s+p^3*r^3*s-10*n^3*p^3+21*n^3*p^2*r-9*n^3*p^2*s -12*n^3*p*r^2+18*n^3*p*r*s+n^3*r^3-9*n^3*r^2*s+36*n^2*p^4-81*n^2*p^3*r+33*n^2*p ^3*s+54*n^2*p^2*r^2-72*n^2*p^2*r*s-9*n^2*p*r^3+45*n^2*p*r^2*s-6*n^2*r^3*s-42*n* p^5+99*n*p^4*r-39*n*p^4*s-72*n*p^3*r^2+90*n*p^3*r*s+15*n*p^2*r^3-63*n*p^2*r^2*s +12*n*p*r^3*s+16*p^6-39*p^5*r+15*p^5*s+30*p^4*r^2-36*p^4*r*s-7*p^3*r^3+27*p^3*r ^2*s-6*p^2*r^3*s+35*n^3*p^2-44*n^3*p*r+26*n^3*p*s+9*n^3*r^2-26*n^3*r*s-165*n^2* p^3+258*n^2*p^2*r-132*n^2*p^2*s-99*n^2*p*r^2+186*n^2*p*r*s+6*n^2*r^3-54*n^2*r^2 *s+237*n*p^4-420*n*p^3*r+198*n*p^3*s+207*n*p^2*r^2-330*n*p^2*r*s-24*n*p*r^3+144 *n*p*r^2*s-12*n*r^3*s-107*p^5+206*p^4*r-92*p^4*s-117*p^3*r^2+170*p^3*r*s+18*p^2 *r^3-90*p^2*r^2*s+12*p*r^3*s-50*n^3*p+26*n^3*r-24*n^3*s+360*n^2*p^2-342*n^2*p*r +228*n^2*p*s+54*n^2*r^2-156*n^2*r*s-690*n*p^3+858*n*p^2*r-492*n*p^2*s-252*n*p*r ^2+528*n*p*r*s+12*n*r^3-108*n*r^2*s+388*p^4-566*p^3*r+296*p^3*s+222*p^2*r^2-396 *p^2*r*s-20*p*r^3+132*p*r^2*s-8*r^3*s+24*n^3-372*n^2*p+156*n^2*r-144*n^2*s+1092 *n*p^2-840*n*p*r+600*n*p*s+108*n*r^2-312*n*r*s-824*p^3+852*p^2*r-528*p^2*s-204* p*r^2+456*p*r*s+8*r^3-72*r^2*s+144*n^2-888*n*p+312*n*r-288*n*s+1024*p^2-664*p*r +496*p*s+72*r^2-208*r*s+288*n-688*p+208*r-192*s+192)/(n^2+2*n+1)/(n+r+1)/(p-r-1 )^3*A[n](r,s)+(3*n^3*p^6-14*n^3*p^5*r+4*n^3*p^5*s+25*n^3*p^4*r^2-20*n^3*p^4*r*s -20*n^3*p^3*r^3+40*n^3*p^3*r^2*s+5*n^3*p^2*r^4-40*n^3*p^2*r^3*s+2*n^3*p*r^5+20* n^3*p*r^4*s-n^3*r^6-4*n^3*r^5*s-6*n^2*p^7+24*n^2*p^6*r-9*n^2*p^6*s-30*n^2*p^5*r ^2+42*n^2*p^5*r*s-75*n^2*p^4*r^2*s+30*n^2*p^3*r^4+60*n^2*p^3*r^3*s-24*n^2*p^2*r ^5-15*n^2*p^2*r^4*s+6*n^2*p*r^6-6*n^2*p*r^5*s+3*n^2*r^6*s+3*n*p^8-6*n*p^7*r+6*n *p^7*s-15*n*p^6*r^2-24*n*p^6*r*s+60*n*p^5*r^3+30*n*p^5*r^2*s-75*n*p^4*r^4+42*n* p^3*r^5-30*n*p^3*r^4*s-9*n*p^2*r^6+24*n*p^2*r^5*s-6*n*p*r^6*s-4*p^8*r-p^8*s+20* p^7*r^2+2*p^7*r*s-40*p^6*r^3+5*p^6*r^2*s+40*p^5*r^4-20*p^5*r^3*s-20*p^4*r^5+25* p^4*r^4*s+4*p^3*r^6-14*p^3*r^5*s+3*p^2*r^6*s-39*n^3*p^5+149*n^3*p^4*r-46*n^3*p^ 4*s-206*n^3*p^3*r^2+184*n^3*p^3*r*s+114*n^3*p^2*r^3-276*n^3*p^2*r^2*s-11*n^3*p* r^4+184*n^3*p*r^3*s-7*n^3*r^5-46*n^3*r^4*s+96*n^2*p^6-330*n^2*p^5*r+129*n^2*p^5 *s+348*n^2*p^4*r^2-507*n^2*p^4*r*s-12*n^2*p^3*r^3+738*n^2*p^3*r^2*s-192*n^2*p^2 *r^4-462*n^2*p^2*r^3*s+102*n^2*p*r^5+93*n^2*p*r^4*s-12*n^2*r^6+9*n^2*r^5*s-60*n *p^7+123*n*p^6*r-105*n*p^6*s+147*n*p^5*r^2+372*n*p^5*r*s-618*n*p^4*r^3-423*n*p^ 4*r^2*s+642*n*p^3*r^4+72*n*p^3*r^3*s-273*n*p^2*r^5+177*n*p^2*r^4*s+39*n*p*r^6-\ 108*n*p*r^5*s+15*n*r^6*s+3*p^8+58*p^7*r+22*p^7*s-289*p^6*r^2-49*p^6*r*s+516*p^5 *r^3-39*p^5*r^2*s-439*p^4*r^4+206*p^4*r^3*s+178*p^3*r^5-224*p^3*r^4*s-27*p^2*r^ 6+99*p^2*r^5*s-15*p*r^6*s+202*n^3*p^4-606*n^3*p^3*r+202*n^3*p^3*s+606*n^3*p^2*r ^2-606*n^3*p^2*r*s-202*n^3*p*r^3+606*n^3*p*r^2*s-202*n^3*r^3*s-636*n^2*p^5+1830 *n^2*p^4*r-744*n^2*p^4*s-1566*n^2*p^3*r^2+2370*n^2*p^3*r*s+78*n^2*p^2*r^3-2646* n^2*p^2*r^2*s+402*n^2*p*r^4+1158*n^2*p*r^3*s-108*n^2*r^5-138*n^2*r^4*s+507*n*p^ 6-1005*n*p^5*r+765*n*p^5*s-486*n*p^4*r^2-2337*n*p^4*r*s+2460*n*p^3*r^3+2304*n*p ^3*r^2*s-2019*n*p^2*r^4-540*n*p^2*r^3*s+585*n*p*r^5-309*n*p*r^4*s-42*n*r^6+117* n*r^5*s-54*p^7-333*p^6*r-204*p^6*s+1731*p^5*r^2+459*p^5*r*s-2716*p^4*r^3+21*p^4 *r^2*s+1902*p^3*r^4-796*p^3*r^3*s-591*p^2*r^5+732*p^2*r^4*s+61*p*r^6-231*p*r^5* s+19*r^6*s-535*n^3*p^3+1178*n^3*p^2*r-427*n^3*p^2*s-751*n^3*p*r^2+854*n^3*p*r*s +108*n^3*r^3-427*n^3*r^2*s+2259*n^2*p^4-5220*n^2*p^3*r+2211*n^2*p^3*s+3387*n^2* p^2*r^2-5352*n^2*p^2*r*s-150*n^2*p*r^3+4071*n^2*p*r^2*s-276*n^2*r^4-930*n^2*r^3 *s-2361*n*p^5+4284*n*p^4*r-3003*n*p^4*s+447*n*p^3*r^2+7590*n*p^3*r*s-4716*n*p^2 *r^3-6033*n*p^2*r^2*s+2760*n*p*r^4+1308*n*p*r^3*s-414*n*r^5+138*n*r^4*s+414*p^6 +919*p^5*r+1042*p^5*s-5543*p^4*r^2-2207*p^4*r*s+7448*p^3*r^3+619*p^3*r^2*s-4059 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s+696)/(n^2*p^2-2*n^2*p*r+n^2*r^2-3*n^2*p+3*n^2*r+2*n*p^2-4*n*p*r+2*n*r^2+2*n^2 -6*n*p+6*n*r+p^2-2*p*r+r^2+4*n-3*p+3*r+2)/(n*p^3-3*n*p^2*r+3*n*p*r^2-n*r^3+p^3* r-3*p^2*r^2+3*p*r^3-r^4-6*n*p^2+12*n*p*r-6*n*r^2+p^3-9*p^2*r+15*p*r^2-7*r^3+11* n*p-11*n*r-6*p^2+23*p*r-17*r^2-6*n+11*p-17*r-6)*diff(diff(A[n](r,s),r),r)+(n^3* p^4+4*n^3*p^3*s-6*n^3*p^2*r^2-12*n^3*p^2*r*s+8*n^3*p*r^3+12*n^3*p*r^2*s-3*n^3*r ^4-4*n^3*r^3*s-6*n^2*p^4*r-3*n^2*p^4*s+12*n^2*p^3*r^2+18*n^2*p^2*r^2*s-12*n^2*p *r^4-24*n^2*p*r^3*s+6*n^2*r^5+9*n^2*r^4*s+9*n*p^4*r^2+6*n*p^4*r*s-24*n*p^3*r^3-\ 12*n*p^3*r^2*s+18*n*p^2*r^4+12*n*p*r^4*s-3*n*r^6-6*n*r^5*s-4*p^4*r^3-3*p^4*r^2* s+12*p^3*r^4+8*p^3*r^3*s-12*p^2*r^5-6*p^2*r^4*s+4*p*r^6+r^6*s-10*n^3*p^3-30*n^3 *p^2*s+30*n^3*p*r^2+60*n^3*p*r*s-20*n^3*r^3-30*n^3*r^2*s+3*n^2*p^4+60*n^2*p^3*r +42*n^2*p^3*s-108*n^2*p^2*r^2-36*n^2*p^2*r*s+24*n^2*p*r^3-54*n^2*p*r^2*s+21*n^2 *r^4+48*n^2*r^3*s-12*n*p^4*r-6*n*p^4*s-66*n*p^3*r^2-60*n*p^3*r*s+180*n*p^2*r^3+ 126*n*p^2*r^2*s-114*n*p*r^4-48*n*p*r^3*s+12*n*r^5-12*n*r^4*s+9*p^4*r^2+6*p^4*r* s+16*p^3*r^3+18*p^3*r^2*s-72*p^2*r^4-60*p^2*r^3*s+60*p*r^5+42*p*r^4*s-13*r^6-6* r^5*s+35*n^3*p^2+70*n^3*p*s-35*n^3*r^2-70*n^3*r*s-30*n^2*p^3-210*n^2*p^2*r-195* n^2*p^2*s+300*n^2*p*r^2+180*n^2*p*r*s-60*n^2*r^3+15*n^2*r^2*s+3*n*p^4+120*n*p^3 *r+72*n*p^3*s+117*n*p^2*r^2+174*n*p^2*r*s-396*n*p*r^3-354*n*p*r^2*s+156*n*r^4+ 108*n*r^3*s-6*p^4*r-3*p^4*s-78*p^3*r^2-60*p^3*r*s+40*p^2*r^3+3*p^2*r^2*s+108*p* r^4+116*p*r^3*s-64*r^5-56*r^4*s-50*n^3*p-50*n^3*s+105*n^2*p^2+300*n^2*p*r+360*n ^2*p*s-255*n^2*r^2-210*n^2*r*s-30*n*p^3-420*n*p^2*r-300*n*p^2*s+60*n*p*r^2-120* n*p*r*s+240*n*r^3+270*n*r^2*s+p^4+60*p^3*r+34*p^3*s+219*p^2*r^2+198*p^2*r*s-212 *p*r^3-138*p*r^2*s-18*r^4-44*r^3*s+24*n^3-150*n^2*p-144*n^2*r-222*n^2*s+105*n*p ^2+600*n*p*r+510*n*p*s-189*n*r^2-66*n*r*s-10*p^3-210*p^2*r-135*p^2*s-210*p*r^2-\ 240*p*r*s+184*r^3+153*r^2*s+72*n^2-150*n*p-288*n*r-294*n*s+35*p^2+300*p*r+220*p *s+31*r^2+74*r*s+72*n-50*p-144*r-122*s+24)/(n^2*p^3-3*n^2*p^2*r+3*n^2*p*r^2-n^2 *r^3-6*n^2*p^2+12*n^2*p*r-6*n^2*r^2+2*n*p^3-6*n*p^2*r+6*n*p*r^2-2*n*r^3+11*n^2* p-11*n^2*r-12*n*p^2+24*n*p*r-12*n*r^2+p^3-3*p^2*r+3*p*r^2-r^3-6*n^2+22*n*p-22*n *r-6*p^2+12*p*r-6*r^2-12*n+11*p-11*r-6)/(n+r+1)*diff(diff(diff(A[n](r,s),r),r), r)-(n^3*r+n^3*s-3*n^2*r^2-3*n^2*r*s+3*n*r^3+3*n*r^2*s-r^4-r^3*s+3*n^2*r+3*n^2*s -6*n*r^2-6*n*r*s+3*r^3+3*r^2*s+3*n*r+3*n*s-3*r^2-3*r*s+r+s)/(n^2+2*n+1)/(n+r+1) *diff(diff(diff(diff(A[n](r,s),r),r),r),r)+(n*p^3-3*n*p^2*r+3*n*p*r^2-n*r^3+p^4 -3*p^3*r+3*p^2*r^2-p*r^3-9*n*p^2+18*n*p*r-9*n*r^2-9*p^3+18*p^2*r-9*p*r^2+26*n*p -26*n*r+26*p^2-26*p*r-24*n-24*p)/(n*p-n*r+p*r-r^2-n+p-2*r-1)/(p-r-1)^2*A[n+1](r ,s)-(4*n*p^5-20*n*p^4*r+40*n*p^3*r^2-40*n*p^2*r^3+20*n*p*r^4-4*n*r^5+3*p^6-14*p ^5*r+25*p^4*r^2-20*p^3*r^3+5*p^2*r^4+2*p*r^5-r^6-46*n*p^4+184*n*p^3*r-276*n*p^2 *r^2+184*n*p*r^3-46*n*r^4-35*p^5+129*p^4*r-166*p^3*r^2+74*p^2*r^3+9*p*r^4-11*r^ 5+202*n*p^3-606*n*p^2*r+606*n*p*r^2-202*n*r^3+156*p^4-422*p^3*r+330*p^2*r^2-18* p*r^3-46*r^4-427*n*p^2+854*n*p*r-427*n*r^2-333*p^3+572*p^2*r-145*p*r^2-94*r^3+ 441*n*p-441*n*r+340*p^2-239*p*r-101*r^2-180*n-125*p-55*r-12)/(p-r-1)/(p^2-2*p*r +r^2-3*p+3*r+2)/(n*p^2-2*n*p*r+n*r^2+p^2*r-2*p*r^2+r^3-3*n*p+3*n*r+p^2-5*p*r+4* r^2+2*n-3*p+5*r+2)*diff(A[n+1](r,s),r)+(6*n*p^5-30*n*p^4*r+60*n*p^3*r^2-60*n*p^ 2*r^3+30*n*p*r^4-6*n*r^5+3*p^6-12*p^5*r+15*p^4*r^2-15*p^2*r^4+12*p*r^5-3*r^6-72 *n*p^4+288*n*p^3*r-432*n*p^2*r^2+288*n*p*r^3-72*n*r^4-36*p^5+108*p^4*r-72*p^3*r ^2-72*p^2*r^3+108*p*r^4-36*r^5+333*n*p^3-999*n*p^2*r+999*n*p*r^2-333*n*r^3+164* p^4-323*p^3*r-15*p^2*r^2+343*p*r^3-169*r^4-744*n*p^2+1488*n*p*r-744*n*r^2-347*p ^3+297*p^2*r+447*p*r^2-397*r^3+805*n*p-805*n*r+313*p^2+179*p*r-492*r^2-340*n-33 *p-307*r-76)/(n*p^3-3*n*p^2*r+3*n*p*r^2-n*r^3+p^3*r-3*p^2*r^2+3*p*r^3-r^4-6*n*p ^2+12*n*p*r-6*n*r^2+p^3-9*p^2*r+15*p*r^2-7*r^3+11*n*p-11*n*r-6*p^2+23*p*r-17*r^ 2-6*n+11*p-17*r-6)/(p^2-2*p*r+r^2-3*p+3*r+2)*diff(diff(A[n+1](r,s),r),r)-(4*n*p ^3-12*n*p^2*r+12*n*p*r^2-4*n*r^3+p^4-6*p^2*r^2+8*p*r^3-3*r^4-30*n*p^2+60*n*p*r-\ 30*n*r^2-6*p^3-12*p^2*r+42*p*r^2-24*r^3+70*n*p-70*n*r+5*p^2+60*p*r-65*r^2-50*n+ 20*p-70*r-26)/(n*p^3-3*n*p^2*r+3*n*p*r^2-n*r^3+p^3*r-3*p^2*r^2+3*p*r^3-r^4-6*n* p^2+12*n*p*r-6*n*r^2+p^3-9*p^2*r+15*p*r^2-7*r^3+11*n*p-11*n*r-6*p^2+23*p*r-17*r ^2-6*n+11*p-17*r-6)*diff(diff(diff(A[n+1](r,s),r),r),r)+diff(diff(diff(diff(A[n +1](r,s),r),r),r),r) = 0 (8*n^6+36*n^5*q-24*n^5*s+66*n^4*q^2-84*n^4*q*s+24*n^4*s^2+63*n^3*q^3-114*n^3*q^ 2*s+60*n^3*q*s^2-8*n^3*s^3+33*n^2*q^4-75*n^2*q^3*s+54*n^2*q^2*s^2-12*n^2*q*s^3+ 9*n*q^5-24*n*q^4*s+21*n*q^3*s^2-6*n*q^2*s^3+q^6-3*q^5*s+3*q^4*s^2-q^3*s^3-4*n^5 -8*n^4*q-4*n^4*s-n^3*q^2-28*n^3*q*s+20*n^3*s^2+7*n^2*q^3-45*n^2*q^2*s+48*n^2*q* s^2-12*n^2*s^3+5*n*q^4-26*n*q^3*s+33*n*q^2*s^2-12*n*q*s^3+q^5-5*q^4*s+7*q^3*s^2 -3*q^2*s^3-26*n^4-71*n^3*q+38*n^3*s-69*n^2*q^2+63*n^2*q*s-6*n^2*s^2-28*n*q^3+30 *n*q^2*s+3*n*q*s^2-6*n*s^3-4*q^4+4*q^3*s+3*q^2*s^2-3*q*s^3-7*n^3-27*n^2*q+33*n^ 2*s-24*n*q^2+42*n*q*s-9*n*s^2-6*q^3+12*q^2*s-3*q*s^2-s^3+16*n^2+11*n*q+10*n*s+q ^2+7*q*s-2*s^2+11*n+5*q+s+2)/(n^2+2*n+1)/(n^3+2*n^2*q-3*n^2*s+n*q^2-4*n*q*s+3*n *s^2-q^2*s+2*q*s^2-s^3+n^2+2*n*q-2*n*s+q^2-2*q*s+s^2)*A[n](r,s)-(12*n^7+60*n^6* q-36*n^6*s+123*n^5*q^2-132*n^5*q*s+24*n^5*s^2+132*n^4*q^3-177*n^4*q^2*s+24*n^4* q*s^2+24*n^4*s^3+78*n^3*q^4-96*n^3*q^3*s-66*n^3*q^2*s^2+120*n^3*q*s^3-36*n^3*s^ 4+24*n^2*q^5-6*n^2*q^4*s-120*n^2*q^3*s^2+174*n^2*q^2*s^3-84*n^2*q*s^4+12*n^2*s^ 5+3*n*q^6+12*n*q^5*s-66*n*q^4*s^2+96*n*q^3*s^3-57*n*q^2*s^4+12*n*q*s^5+3*q^6*s-\ 12*q^5*s^2+18*q^4*s^3-12*q^3*s^4+3*q^2*s^5-6*n^6-9*n^5*q-18*n^5*s+21*n^4*q^2-\ 129*n^4*q*s+84*n^4*s^2+60*n^3*q^3-276*n^3*q^2*s+294*n^3*q*s^2-84*n^3*s^3+54*n^2 *q^4-252*n^2*q^3*s+342*n^2*q^2*s^2-162*n^2*q*s^3+18*n^2*s^4+21*n*q^5-102*n*q^4* s+156*n*q^3*s^2-84*n*q^2*s^3+3*n*q*s^4+6*n*s^5+3*q^6-15*q^5*s+24*q^4*s^2-12*q^3 *s^3-3*q^2*s^4+3*q*s^5-41*n^5-145*n^4*q+85*n^4*s-194*n^3*q^2+196*n^3*q*s-26*n^3 *s^2-120*n^2*q^3+138*n^2*q^2*s+18*n^2*q*s^2-38*n^2*s^3-33*n*q^4+24*n*q^3*s+66*n *q^2*s^2-76*n*q*s^3+19*n*s^4-3*q^5-3*q^4*s+24*q^3*s^2-26*q^2*s^3+7*q*s^4+s^5-4* n^4-40*n^3*q+64*n^3*s-72*n^2*q^2+168*n^2*q*s-72*n^2*s^2-45*n*q^3+126*n*q^2*s-84 *n*q*s^2+8*n*s^3-9*q^4+27*q^3*s-18*q^2*s^2-4*q*s^3+4*s^4+28*n^3+54*n^2*q-24*n^2 *s+27*n*q^2-24*n*s^2+3*q^3+9*q^2*s-18*q*s^2+4*s^3+4*n^2+17*n*q-26*n*s+7*q^2-11* q*s-2*s^2-7*n-q-5*s-2)/(n^3+n^2*q-n^2*s+2*n^2+2*n*q-2*n*s+n+q-s)/(n^2+n*q-2*n*s -q*s+s^2+n+q-s)/(-s-1+n+q)*diff(A[n](r,s),s)+(6*n^6+21*n^5*q-6*n^5*s+27*n^4*q^2 -3*n^4*q*s-12*n^4*s^2+15*n^3*q^3+18*n^3*q^2*s-42*n^3*q*s^2+12*n^3*s^3+3*n^2*q^4 +21*n^2*q^3*s-36*n^2*q^2*s^2+6*n^2*q*s^3+6*n^2*s^4+6*n*q^4*s-3*n*q^3*s^2-18*n*q ^2*s^3+21*n*q*s^4-6*n*s^5+3*q^4*s^2-9*q^3*s^3+9*q^2*s^4-3*q*s^5+6*n^5+27*n^4*q-\ 24*n^4*s+42*n^3*q^2-60*n^3*q*s+12*n^3*s^2+27*n^2*q^3-36*n^2*q^2*s-18*n^2*q*s^2+ 24*n^2*s^3+6*n*q^4+6*n*q^3*s-54*n*q^2*s^2+60*n*q*s^3-18*n*s^4+6*q^4*s-21*q^3*s^ 2+24*q^2*s^3-9*q*s^4-16*n^4-25*n^3*q-14*n^3*s-3*n^2*q^2-63*n^2*q*s+42*n^2*s^2+9 *n*q^3-60*n*q^2*s+57*n*q*s^2-10*n*s^3+3*q^4-15*q^3*s+15*q^2*s^2-q*s^3-2*s^4-22* n^3-45*n^2*q+24*n^2*s-24*n*q^2+6*n*q*s+18*n*s^2-3*q^3-6*q^2*s+15*q*s^2-4*s^3-12 *n*q+24*n*s-6*q^2+12*q*s+8*n+2*q+4*s+2)/(n^4+2*n^3*q-2*n^3*s+n^2*q^2-2*n^2*q*s+ n^2*s^2+n^3+3*n^2*q-3*n^2*s+2*n*q^2-4*n*q*s+2*n*s^2-n^2+q^2-2*q*s+s^2-n-q+s)/(n -s+1)*diff(diff(A[n](r,s),s),s)-(n^4+n^3*q+2*n^3*s+3*n^2*q*s+3*n*q*s^2-2*n*s^3+ q*s^3-s^4+4*n^3+3*n^2*q+6*n^2*s+6*n*q*s+3*q*s^2-2*s^3+6*n^2+3*n*q+6*n*s+3*q*s+4 *n+q+2*s+1)/(n^2+2*n+1)/(n-s+1)*diff(diff(diff(A[n](r,s),s),s),s)-q*(n^3+3*n^2* q-3*n^2*s+3*n*q^2-6*n*q*s+3*n*s^2+q^3-3*q^2*s+3*q*s^2-s^3-3*n^2-6*n*q+6*n*s-3*q ^2+6*q*s-3*s^2+2*n+2*q-2*s)/(n^2+n*q-2*n*s-q*s+s^2+2*n+q-2*s+1)/(n^2+2*n*q-2*n* s+q^2-2*q*s+s^2+2*n+2*q-2*s+1)*A[n+1](r,s)-(n^6+2*n^5*q-6*n^5*s-5*n^4*q^2-10*n^ 4*q*s+15*n^4*s^2-20*n^3*q^3+20*n^3*q^2*s+20*n^3*q*s^2-20*n^3*s^3-25*n^2*q^4+60* n^2*q^3*s-30*n^2*q^2*s^2-20*n^2*q*s^3+15*n^2*s^4-14*n*q^5+50*n*q^4*s-60*n*q^3*s ^2+20*n*q^2*s^3+10*n*q*s^4-6*n*s^5-3*q^6+14*q^5*s-25*q^4*s^2+20*q^3*s^3-5*q^2*s ^4-2*q*s^5+s^6+n^5+11*n^4*q-5*n^4*s+34*n^3*q^2-44*n^3*q*s+10*n^3*s^2+46*n^2*q^3 -102*n^2*q^2*s+66*n^2*q*s^2-10*n^2*s^3+29*n*q^4-92*n*q^3*s+102*n*q^2*s^2-44*n*q *s^3+5*n*s^4+7*q^5-29*q^4*s+46*q^3*s^2-34*q^2*s^3+11*q*s^4-s^5-4*n^4-10*n^3*q+ 16*n^3*s-6*n^2*q^2+30*n^2*q*s-24*n^2*s^2+2*n*q^3+12*n*q^2*s-30*n*q*s^2+16*n*s^3 +2*q^4-2*q^3*s-6*q^2*s^2+10*q*s^3-4*s^4-6*n^3-19*n^2*q+18*n^2*s-20*n*q^2+38*n*q *s-18*n*s^2-7*q^3+20*q^2*s-19*q*s^2+6*s^3+n^2-3*n*q-2*n*s-4*q^2+3*q*s+s^2+5*n+3 *q-5*s+2)/(n^6+5*n^5*q-6*n^5*s+10*n^4*q^2-25*n^4*q*s+15*n^4*s^2+10*n^3*q^3-40*n ^3*q^2*s+50*n^3*q*s^2-20*n^3*s^3+5*n^2*q^4-30*n^2*q^3*s+60*n^2*q^2*s^2-50*n^2*q *s^3+15*n^2*s^4+n*q^5-10*n*q^4*s+30*n*q^3*s^2-40*n*q^2*s^3+25*n*q*s^4-6*n*s^5-q ^5*s+5*q^4*s^2-10*q^3*s^3+10*q^2*s^4-5*q*s^5+s^6+4*n^5+17*n^4*q-20*n^4*s+28*n^3 *q^2-68*n^3*q*s+40*n^3*s^2+22*n^2*q^3-84*n^2*q^2*s+102*n^2*q*s^2-40*n^2*s^3+8*n *q^4-44*n*q^3*s+84*n*q^2*s^2-68*n*q*s^3+20*n*s^4+q^5-8*q^4*s+22*q^3*s^2-28*q^2* s^3+17*q*s^4-4*s^5+6*n^4+21*n^3*q-24*n^3*s+27*n^2*q^2-63*n^2*q*s+36*n^2*s^2+15* n*q^3-54*n*q^2*s+63*n*q*s^2-24*n*s^3+3*q^4-15*q^3*s+27*q^2*s^2-21*q*s^3+6*s^4+4 *n^3+11*n^2*q-12*n^2*s+10*n*q^2-22*n*q*s+12*n*s^2+3*q^3-10*q^2*s+11*q*s^2-4*s^3 +n^2+2*n*q-2*n*s+q^2-2*q*s+s^2)*diff(A[n+1](r,s),s)+(3*n^6+12*n^5*q-18*n^5*s+15 *n^4*q^2-60*n^4*q*s+45*n^4*s^2-60*n^3*q^2*s+120*n^3*q*s^2-60*n^3*s^3-15*n^2*q^4 +90*n^2*q^2*s^2-120*n^2*q*s^3+45*n^2*s^4-12*n*q^5+30*n*q^4*s-60*n*q^2*s^3+60*n* q*s^4-18*n*s^5-3*q^6+12*q^5*s-15*q^4*s^2+15*q^2*s^4-12*q*s^5+3*s^6+12*n^4*q+48* n^3*q^2-48*n^3*q*s+72*n^2*q^3-144*n^2*q^2*s+72*n^2*q*s^2+48*n*q^4-144*n*q^3*s+ 144*n*q^2*s^2-48*n*q*s^3+12*q^5-48*q^4*s+72*q^3*s^2-48*q^2*s^3+12*q*s^4-11*n^4-\ 41*n^3*q+44*n^3*s-57*n^2*q^2+123*n^2*q*s-66*n^2*s^2-35*n*q^3+114*n*q^2*s-123*n* q*s^2+44*n*s^3-8*q^4+35*q^3*s-57*q^2*s^2+41*q*s^3-11*s^4-5*n^3-21*n^2*q+15*n^2* s-27*n*q^2+42*n*q*s-15*n*s^2-11*q^3+27*q^2*s-21*q*s^2+5*s^3+6*n^2+11*n*q-12*n*s +5*q^2-11*q*s+6*s^2+n+3*q-s-2)/(-s-1+n+q)/(n^5+4*n^4*q-5*n^4*s+6*n^3*q^2-16*n^3 *q*s+10*n^3*s^2+4*n^2*q^3-18*n^2*q^2*s+24*n^2*q*s^2-10*n^2*s^3+n*q^4-8*n*q^3*s+ 18*n*q^2*s^2-16*n*q*s^3+5*n*s^4-q^4*s+4*q^3*s^2-6*q^2*s^3+4*q*s^4-s^5+3*n^4+10* n^3*q-12*n^3*s+12*n^2*q^2-30*n^2*q*s+18*n^2*s^2+6*n*q^3-24*n*q^2*s+30*n*q*s^2-\ 12*n*s^3+q^4-6*q^3*s+12*q^2*s^2-10*q*s^3+3*s^4+3*n^3+8*n^2*q-9*n^2*s+7*n*q^2-16 *n*q*s+9*n*s^2+2*q^3-7*q^2*s+8*q*s^2-3*s^3+n^2+2*n*q-2*n*s+q^2-2*q*s+s^2)*diff( diff(A[n+1](r,s),s),s)-(3*n^4+8*n^3*q-12*n^3*s+6*n^2*q^2-24*n^2*q*s+18*n^2*s^2-\ 12*n*q^2*s+24*n*q*s^2-12*n*s^3-q^4+6*q^2*s^2-8*q*s^3+3*s^4+6*n^2*q+12*n*q^2-12* n*q*s+6*q^3-12*q^2*s+6*q*s^2-7*n^2-12*n*q+14*n*s-5*q^2+12*q*s-7*s^2-2*n-4*q+2*s +2)/(n+q-s)/(n^2+n*q-2*n*s-q*s+s^2+2*n+q-2*s+1)/(-s-1+n+q)*diff(diff(diff(A[n+1 ](r,s),s),s),s)+diff(diff(diff(diff(A[n+1](r,s),s),s),s),s) = 0 ------------------------------------------------- This took, 0.735, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ A[n](r, s) = ) / ----- k = 0 2 (k - 1 + p) (n - k + q) k binomial(n, k) binomial(n + k, k) (r + k) (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)^2*binomial(n+k,k)*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x^k ,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 2 2 2 2 2 3 2 2 2 - (n p - n p r + n p s - n r s - n p + n p r - n p s + n p r s - 3 n p 2 2 2 3 2 2 + n r - 2 n s + 6 n p - 4 n p r + 5 n p s - 3 n r s - p + p r - p s 2 2 + p r s + 2 n - 11 n p + 3 n r - 6 n s + 5 p - 3 p r + 4 p s - 2 r s 2 2 + 6 n - 8 p + 2 r - 4 s + 4) A[n](r, s)/((n + r + 1) %1) + (n p 2 2 2 2 2 2 2 2 + 2 n p s - n r - 2 n r s - 2 n p r - n p s + 2 n p r + n r s 2 2 2 2 2 - 3 n p - 3 n s + 2 n p + 6 n p r + 7 n p s - 5 n r - 4 n r s - 2 p r 2 2 2 2 2 - p s + 2 p r + r s + 2 n - 6 n p - 4 n r - 8 n s + p + 6 p r + 5 p s 2 /d \ / - 4 r - 2 r s + 4 n - 3 p - 4 r - 5 s + 2) |-- A[n](r, s)| / ( \dr / / 2 (p - r - 1) (n + r + 1) ) - 2 2 2 2 (n r + n s - n r - n r s + 2 n r + 2 n s - r - r s + r + s) / 2 \ |d | / 2 |--- A[n](r, s)| / (n + r + 1) + | 2 | / \dr / 2 2 2 3 2 2 2 (n p - n r + 2 n p - 2 n p r + p - p r - 2 n - 4 n p - 2 p ) A[n + 1](r, s)/((n + r + 1) %1) 2 /d \ (2 n p - 2 n r + 2 p - 2 p r - 3 n - 4 p + r + 1) |-- A[n + 1](r, s)| \dr / - ---------------------------------------------------------------------- %1 / 2 \ |d | + |--- A[n + 1](r, s)| = 0 | 2 | \dr / 2 %1 := n p - n r + p r - r - n + p - 2 r - 1 4 3 3 2 2 2 2 2 3 2 (2 n + 5 n q - 4 n s + 4 n q - 6 n q s + 2 n s + n q - 2 n q s 2 3 2 2 2 2 3 + n q s + 7 n + 13 n q - 10 n s + 7 n q - 10 n q s + 3 n s + q 2 2 2 2 2 - 2 q s + q s + 9 n + 11 n q - 8 n s + 3 q - 4 q s + s + 5 n + 3 q 4 3 3 2 2 - 2 s + 1) A[n](r, s)/((n - s + 1) %1) - (n + 2 n q - n s + n q 2 2 2 2 3 3 2 2 2 - n s + n q s - 2 n q s + n s + 4 n + 6 n q - 3 n s + 2 n q 2 2 2 3 2 2 2 - 2 n s + q s - 2 q s + s + 6 n + 6 n q - 3 n s + q - s + 4 n + 2 q 2 /d \ (n + q - s) q A[n + 1](r, s) - s + 1) |-- A[n](r, s)|/((n - s + 1) %1) + ----------------------------- \ds / (n - s + 1) %1 /d \ + |-- A[n + 1](r, s)| \ds / / 2 \ 2 2 |d | (n + n q - 2 n s - q s + s + 2 n + q - 2 s + 1) |--- A[n + 1](r, s)| | 2 | \ds / + ---------------------------------------------------------------------- %1 = 0 2 %1 := 2 n q + 2 q - 2 q s - n + s - 1 and in Maple notation -(n^2*p^2-n^2*p*r+n^2*p*s-n^2*r*s-n*p^3+n*p^2*r-n*p^2*s+n*p*r*s-3*n^2*p+n^2*r-2 *n^2*s+6*n*p^2-4*n*p*r+5*n*p*s-3*n*r*s-p^3+p^2*r-p^2*s+p*r*s+2*n^2-11*n*p+3*n*r -6*n*s+5*p^2-3*p*r+4*p*s-2*r*s+6*n-8*p+2*r-4*s+4)/(n+r+1)/(n*p-n*r+p*r-r^2-n+p-\ 2*r-1)*A[n](r,s)+(n^2*p^2+2*n^2*p*s-n^2*r^2-2*n^2*r*s-2*n*p^2*r-n*p^2*s+2*n*p*r ^2+n*r^2*s-3*n^2*p-3*n^2*s+2*n*p^2+6*n*p*r+7*n*p*s-5*n*r^2-4*n*r*s-2*p^2*r-p^2* s+2*p*r^2+r^2*s+2*n^2-6*n*p-4*n*r-8*n*s+p^2+6*p*r+5*p*s-4*r^2-2*r*s+4*n-3*p-4*r -5*s+2)/(p-r-1)/(n+r+1)^2*diff(A[n](r,s),r)-(n^2*r+n^2*s-n*r^2-n*r*s+2*n*r+2*n* s-r^2-r*s+r+s)/(n+r+1)^2*diff(diff(A[n](r,s),r),r)+(n^2*p-n^2*r+2*n*p^2-2*n*p*r +p^3-p^2*r-2*n^2-4*n*p-2*p^2)/(n+r+1)/(n*p-n*r+p*r-r^2-n+p-2*r-1)*A[n+1](r,s)-( 2*n*p-2*n*r+2*p^2-2*p*r-3*n-4*p+r+1)/(n*p-n*r+p*r-r^2-n+p-2*r-1)*diff(A[n+1](r, s),r)+diff(diff(A[n+1](r,s),r),r) = 0 (2*n^4+5*n^3*q-4*n^3*s+4*n^2*q^2-6*n^2*q*s+2*n^2*s^2+n*q^3-2*n*q^2*s+n*q*s^2+7* n^3+13*n^2*q-10*n^2*s+7*n*q^2-10*n*q*s+3*n*s^2+q^3-2*q^2*s+q*s^2+9*n^2+11*n*q-8 *n*s+3*q^2-4*q*s+s^2+5*n+3*q-2*s+1)/(n-s+1)/(2*n*q+2*q^2-2*q*s-n+s-1)*A[n](r,s) -(n^4+2*n^3*q-n^3*s+n^2*q^2-n^2*s^2+n*q^2*s-2*n*q*s^2+n*s^3+4*n^3+6*n^2*q-3*n^2 *s+2*n*q^2-2*n*s^2+q^2*s-2*q*s^2+s^3+6*n^2+6*n*q-3*n*s+q^2-s^2+4*n+2*q-s+1)/(n- s+1)/(2*n*q+2*q^2-2*q*s-n+s-1)*diff(A[n](r,s),s)+(n+q-s)*q^2/(n-s+1)/(2*n*q+2*q ^2-2*q*s-n+s-1)*A[n+1](r,s)+diff(A[n+1](r,s),s)+(n^2+n*q-2*n*s-q*s+s^2+2*n+q-2* s+1)/(2*n*q+2*q^2-2*q*s-n+s-1)*diff(diff(A[n+1](r,s),s),s) = 0 ------------------------------------------------- This took, 0.054, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ 2 2 (k - 1 + p) A[n](r, s) = ) binomial(n, k) binomial(n + k, k) (r + k) / ----- k = 0 (n - k + q) k (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)^2*binomial(n+k,k)^2*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x ^k,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 2 3 2 2 2 2 2 2 2 2 2 4 (n p - 2 n p r + n p s + n p r - 2 n p r s + n r s - 2 n p 3 3 2 2 2 2 5 4 + 4 n p r - 2 n p s - 2 n p r + 4 n p r s - 2 n p r s + p - 2 p r 4 3 2 3 2 2 2 2 2 2 + p s + p r - 2 p r s + p r s - 5 n p + 6 n p r - 4 n p s 2 2 2 3 2 2 2 - n r + 4 n r s + 14 n p - 20 n p r + 12 n p s + 6 n p r 2 4 3 3 2 2 2 - 16 n p r s + 4 n r s - 9 p + 14 p r - 8 p s - 5 p r + 12 p r s 2 2 2 2 2 - 4 p r s + 8 n p - 4 n r + 4 n s - 36 n p + 32 n p r - 24 n p s 2 3 2 2 2 - 4 n r + 16 n r s + 32 p - 36 p r + 24 p s + 8 p r - 24 p r s 2 2 2 + 4 r s - 4 n + 40 n p - 16 n r + 16 n s - 56 p + 40 p r - 32 p s 2 - 4 r + 16 r s - 16 n + 48 p - 16 r + 16 s - 16) A[n](r, s)/(%2) - ( 2 3 2 2 2 2 2 2 3 2 2 4 2 n p - 3 n p r + 3 n p s - 6 n p r s + n r + 3 n r s - 2 n p 3 2 2 2 3 3 4 4 - 4 n p s + 6 n p r + 6 n p r s - 4 n p r - 2 n r s + 3 p r + p s 3 2 2 3 2 2 3 2 2 2 2 - 6 p r + 3 p r - 3 p r s + 2 p r s - 9 n p + 9 n p r - 9 n p s 2 3 2 2 2 + 9 n r s + 16 n p - 6 n p r + 24 n p s - 18 n p r - 30 n p r s 3 2 4 3 3 2 2 2 + 8 n r + 6 n r s - 2 p - 18 p r - 10 p s + 33 p r + 6 p r s 3 2 3 2 2 2 2 - 13 p r + 9 p r s - 5 r s + 13 n p - 6 n r + 7 n s - 44 n p 2 3 2 2 + 18 n p r - 44 n p s + 12 n r + 30 n r s + 14 p + 36 p r + 34 p s 2 3 2 2 - 57 p r - 24 p r s + 14 r - 3 r s - 6 n + 50 n p - 12 n r + 26 n s 2 2 - 35 p - 27 p r - 47 p s + 30 r + 21 r s - 20 n + 37 p + 6 r + 23 s - 14 /d \ 2 4 2 3 2 3 2 2 2 ) |-- A[n](r, s)|/(%2) + (n p - n p r + 3 n p s - 3 n p r \dr / 2 2 2 3 2 2 2 4 2 3 4 - 9 n p r s + 5 n p r + 9 n p r s - 2 n r - 3 n r s - 4 n p r 4 3 2 3 2 3 2 2 4 - 2 n p s + 10 n p r + 2 n p r s - 6 n p r + 6 n p r s - 2 n p r 3 5 4 4 2 4 3 3 - 10 n p r s + 2 n r + 4 n r s + 3 p r + 2 p r s - 9 p r 3 2 2 4 2 3 5 4 5 2 3 - 5 p r s + 9 p r + 3 p r s - 3 p r + p r s - r s - 7 n p 2 2 2 2 2 2 2 2 3 2 2 + 6 n p r - 15 n p s + 9 n p r + 30 n p r s - 8 n r - 15 n r s 4 3 3 2 2 2 3 + 2 n p + 26 n p r + 20 n p s - 60 n p r - 30 n p r s + 34 n p r 4 3 4 4 3 2 3 2 3 - 2 n r + 10 n r s - 4 p r - 2 p s - 11 p r - 12 p r s + 42 p r 2 2 4 3 5 4 2 2 2 + 33 p r s - 35 p r - 22 p r s + 8 r + 3 r s + 17 n p - 11 n p r 2 2 2 2 3 2 2 + 23 n p s - 6 n r - 23 n r s - 14 n p - 56 n p r - 64 n p s 2 3 2 4 3 3 + 108 n p r + 82 n p r s - 38 n r - 18 n r s + p + 27 p r + 17 p s 2 2 2 3 2 4 3 2 - 6 p r + 13 p r s - 50 p r - 54 p r s + 28 r + 24 r s - 17 n p 2 2 2 2 + 6 n r - 11 n s + 34 n p + 46 n p r + 80 n p s - 58 n r - 58 n r s 3 2 2 2 3 2 2 - 7 p - 62 p r - 49 p s + 48 p r + 18 p r s + 10 r + 20 r s + 6 n 2 2 - 34 n p - 12 n r - 34 n s + 17 p + 57 p r + 57 p s - 34 r - 23 r s / 2 \ |d | + 12 n - 17 p - 18 r - 23 s + 6) |--- A[n](r, s)|/((-3 + p - r) %2) - ( | 2 | \dr / 2 3 2 3 2 2 2 2 2 2 3 2 2 n p r + n p s - 3 n p r - 3 n p r s + 3 n p r + 3 n p r s 2 4 2 3 3 2 3 2 3 2 2 - n r - n r s - 2 n p r - 2 n p r s + 6 n p r + 6 n p r s 4 3 5 4 3 3 3 2 2 4 - 6 n p r - 6 n p r s + 2 n r + 2 n r s + p r + p r s - 3 p r 2 3 5 4 6 5 2 2 2 2 - 3 p r s + 3 p r + 3 p r s - r - r s - 4 n p r - 4 n p s 2 2 2 2 3 2 2 3 3 + 8 n p r + 8 n p r s - 4 n r - 4 n r s + 2 n p r + 2 n p s 2 2 2 3 2 4 3 + 2 n p r + 2 n p r s - 10 n p r - 10 n p r s + 6 n r + 6 n r s 3 2 3 2 3 2 2 4 3 5 - 2 p r - 2 p r s + 2 p r + 2 p r s + 2 p r + 2 p r s - 2 r 4 2 2 2 2 2 2 2 - 2 r s + 5 n p r + 5 n p s - 5 n r - 5 n r s - 8 n p r - 8 n p s 2 3 2 3 3 2 2 + 6 n p r + 6 n p r s + 2 n r + 2 n r s + p r + p s + 5 p r 2 3 2 4 3 2 2 + 5 p r s - 8 p r - 8 p r s + 2 r + 2 r s - 2 n r - 2 n s 2 2 2 2 + 10 n p r + 10 n p s - 6 n r - 6 n r s - 4 p r - 4 p s - 2 p r 3 2 2 - 2 p r s + 4 r + 4 r s - 4 n r - 4 n s + 5 p r + 5 p s - r - r s - 2 r / 3 \ |d | 2 2 2 2 2 3 - 2 s) |--- A[n](r, s)|/(%1) - (n p - 2 n p r + n r + 2 n p | 3 | \dr / 2 2 4 3 2 2 2 2 2 - 4 n p r + 2 n p r + p - 2 p r + p r - 4 n p + 4 n r - 8 n p 3 2 2 2 + 8 n p r - 4 p + 4 p r + 4 n + 8 n p + 4 p ) A[n + 1](r, s)/(%2) /d \ 2 3 2 2 2 2 2 3 + |-- A[n + 1](r, s)| - (3 n p - 9 n p r + 9 n p r - 3 n r \dr / 4 3 2 2 3 4 4 3 2 + 2 n p - 2 n p r - 6 n p r + 10 n p r - 4 n r + 2 p r - 5 p r 2 3 4 5 2 2 2 2 2 3 + 3 p r + p r - r - 15 n p + 30 n p r - 15 n r - 8 n p 2 2 3 4 3 2 2 3 - 6 n p r + 36 n p r - 22 n r + 2 p - 16 p r + 21 p r - 2 p r 4 2 2 2 2 3 2 - 5 r + 23 n p - 23 n r + 4 n p + 38 n p r - 42 n r - 11 p + 37 p r 2 3 2 2 2 - 18 p r - 8 r - 11 n + 12 n p - 34 n r + 19 p - 26 p r - 4 r - 10 n / 2 \ |d | 2 3 2 2 2 2 - 11 p + r + 1) |--- A[n + 1](r, s)|/(%1) + (n p - 3 n p r + 3 n p r | 2 | \dr / 2 3 3 2 2 3 4 3 2 2 3 - n r + 2 n p r - 6 n p r + 6 n p r - 2 n r + p r - 3 p r 4 5 2 2 2 2 2 3 2 + 3 p r - r - 4 n p + 8 n p r - 4 n r + 2 n p - 14 n p r 2 3 3 2 2 3 4 2 + 22 n p r - 10 n r + 2 p r - 10 p r + 14 p r - 6 r + 5 n p 2 2 2 3 2 2 3 - 5 n r - 8 n p + 26 n p r - 18 n r + p - 11 p r + 24 p r - 14 r 2 2 2 - 2 n + 10 n p - 14 n r - 4 p + 18 p r - 16 r - 4 n + 5 p - 9 r - 2) / 3 \ |d | |--- A[n + 1](r, s)|/(%1) = 0 | 3 | \dr / 2 3 2 2 2 2 2 3 4 3 %1 := 3 n p - 9 n p r + 9 n p r - 3 n r + 4 n p - 10 n p r 2 2 3 4 5 4 3 2 2 3 4 + 6 n p r + 2 n p r - 2 n r + p - p r - 3 p r + 5 p r - 2 p r 2 2 2 2 2 3 2 3 4 - 18 n p + 36 n p r - 18 n r - 24 n p + 36 n p r - 12 n r - 5 p 3 2 2 3 4 2 2 2 - 4 p r + 24 p r - 16 p r + r + 34 n p - 34 n r + 44 n p 2 3 2 2 3 2 - 20 n p r - 24 n r + 4 p + 32 p r - 42 p r + 6 r - 21 n - 22 n p 2 2 - 20 n r + 11 p - 44 p r + 12 r - 6 n - 16 p + 10 r + 3 2 2 2 2 2 3 2 3 4 2 2 %2 := 3 n p - 6 n p r + 3 n r + 4 n p - 6 n p r + 2 n r + p - 3 p r 3 2 2 2 2 3 2 + 2 p r - 9 n p + 9 n r - 12 n p + 6 n p r + 6 n r - 2 p - 6 p r 2 3 2 2 2 + 9 p r - r + 7 n + 8 n p + 6 n r - 2 p + 12 p r - 3 r + 2 n + 5 p - 3 r - 1 5 4 4 3 2 3 3 2 2 3 (4 n + 16 n q - 12 n s + 25 n q - 36 n q s + 12 n s + 19 n q 2 2 2 2 2 3 4 3 2 2 - 39 n q s + 24 n q s - 4 n s + 7 n q - 18 n q s + 15 n q s 3 5 4 3 2 2 3 4 3 3 - 4 n q s + q - 3 q s + 3 q s - q s + 8 n + 26 n q - 20 n s 2 2 2 2 2 3 2 2 + 31 n q - 46 n q s + 16 n s + 16 n q - 34 n q s + 22 n q s 3 4 3 2 2 3 3 2 2 - 4 n s + 3 q - 8 q s + 7 q s - 2 q s + n + 5 n q - 7 n s 2 2 3 2 2 3 2 + 6 n q - 14 n q s + 7 n s + 2 q - 6 q s + 5 q s - s - 7 n - 8 n q 2 2 + 2 n s - 2 q + s - 5 n - 3 q + s - 1) A[n](r, s)/((n - s + 1) %1) - ( 5 4 4 3 2 3 2 3 2 2 4 n + 14 n q - 8 n s + 18 n q - 16 n q s + 10 n q - 6 n q s 2 2 2 3 4 3 2 2 3 - 12 n q s + 8 n s + 2 n q + 4 n q s - 18 n q s + 16 n q s 4 4 3 2 2 3 4 4 3 3 - 4 n s + 2 q s - 6 q s + 6 q s - 2 q s + 11 n + 32 n q - 20 n s 2 2 2 2 2 3 2 3 4 + 33 n q - 36 n q s + 6 n s + 14 n q - 18 n q s + 4 n s + 2 q 3 2 2 3 4 3 2 2 2 - 2 q s - 3 q s + 4 q s - s + 8 n + 20 n q - 16 n s + 16 n q 2 3 2 2 2 2 - 24 n q s + 8 n s + 4 q - 8 q s + 4 q s - 2 n - 4 n s + q - 4 q s 2 /d \ 5 4 + 2 s - 4 n - 2 q - 1) |-- A[n](r, s)|/((n - s + 1) %1) + (n + 3 n q \ds / 4 3 2 3 2 2 3 2 2 2 2 2 3 - n s + 3 n q - 2 n s + n q + 3 n q s - 6 n q s + 2 n s 3 2 2 4 3 2 2 3 4 5 4 + 2 n q s - 3 n q s + n s + q s - 3 q s + 3 q s - s + 4 n 3 3 2 2 2 2 2 3 2 + 10 n q - 4 n s + 8 n q - 2 n q s - 4 n s + 2 n q + 4 n q s 2 3 3 2 2 3 3 2 2 - 10 n q s + 4 n s + 2 q s - 4 q s + 2 q s + 6 n + 12 n q - 6 n s 2 2 3 2 2 3 2 + 7 n q - 4 n q s - 2 n s + q + q s - 4 q s + 2 s + 4 n + 6 n q / 2 \ 2 |d | 2 - 4 n s + 2 q - 2 q s + n + q - s) |--- A[n](r, s)|/((n - s + 1) %1) + q | 2 | \ds / 3 2 2 2 2 3 2 2 (n + 3 n q - 3 n s + 3 n q - 6 n q s + 3 n s + q - 3 q s + 3 q s 3 2 2 2 - s - n - 2 n q + 2 n s - q + 2 q s - s ) A[n + 1](r, s)/( 4 3 2 3 2 3 (n + q - s + 1) (n - s + 1) %1) + (2 n q + 5 n q - 8 n q s + 3 n q 2 2 2 2 4 3 2 2 3 5 - 15 n q s + 12 n q s - n q - 6 n q s + 15 n q s - 8 n q s - q 4 3 2 2 3 4 4 3 2 2 2 2 + q s + 3 q s - 5 q s + 2 q s - n + 4 n s + 6 n q - 6 n s 3 2 3 4 3 2 2 4 3 + 8 n q - 12 n q s + 4 n s + 3 q - 8 q s + 6 q s - s - 2 n 2 2 2 2 2 2 3 - 6 n q + 6 n s - 4 n q + 12 n q s - 6 n s + 4 q s - 6 q s + 2 s 2 /d \ - 4 n q - 3 q + 4 q s + 2 n - 2 s + 1) |-- A[n + 1](r, s)|/( \ds / / 2 \ |d | 3 2 (n + q - s + 1) (n - s + 1) %1) + |--- A[n + 1](r, s)| - (n + 2 n q | 2 | \ds / 2 2 2 2 2 3 2 - 3 n s + n q - 4 n q s + 3 n s - q s + 2 q s - s + 2 n + 3 n q / 3 \ 2 2 |d | - 4 n s + q - 3 q s + 2 s + n + q - s) |--- A[n + 1](r, s)|/(%1) = 0 | 3 | \ds / 3 2 2 2 3 2 3 2 %1 := n - 3 n s - 3 n q + 3 n s - 2 q + 3 q s - s + 3 n + 6 n q - 6 n s 2 2 + 3 q - 6 q s + 3 s + n + 2 q - s - 1 and in Maple notation (n^2*p^3-2*n^2*p^2*r+n^2*p^2*s+n^2*p*r^2-2*n^2*p*r*s+n^2*r^2*s-2*n*p^4+4*n*p^3* r-2*n*p^3*s-2*n*p^2*r^2+4*n*p^2*r*s-2*n*p*r^2*s+p^5-2*p^4*r+p^4*s+p^3*r^2-2*p^3 *r*s+p^2*r^2*s-5*n^2*p^2+6*n^2*p*r-4*n^2*p*s-n^2*r^2+4*n^2*r*s+14*n*p^3-20*n*p^ 2*r+12*n*p^2*s+6*n*p*r^2-16*n*p*r*s+4*n*r^2*s-9*p^4+14*p^3*r-8*p^3*s-5*p^2*r^2+ 12*p^2*r*s-4*p*r^2*s+8*n^2*p-4*n^2*r+4*n^2*s-36*n*p^2+32*n*p*r-24*n*p*s-4*n*r^2 +16*n*r*s+32*p^3-36*p^2*r+24*p^2*s+8*p*r^2-24*p*r*s+4*r^2*s-4*n^2+40*n*p-16*n*r +16*n*s-56*p^2+40*p*r-32*p*s-4*r^2+16*r*s-16*n+48*p-16*r+16*s-16)/(3*n^2*p^2-6* n^2*p*r+3*n^2*r^2+4*n*p^3-6*n*p^2*r+2*n*r^3+p^4-3*p^2*r^2+2*p*r^3-9*n^2*p+9*n^2 *r-12*n*p^2+6*n*p*r+6*n*r^2-2*p^3-6*p^2*r+9*p*r^2-r^3+7*n^2+8*n*p+6*n*r-2*p^2+ 12*p*r-3*r^2+2*n+5*p-3*r-1)*A[n](r,s)-(2*n^2*p^3-3*n^2*p^2*r+3*n^2*p^2*s-6*n^2* p*r*s+n^2*r^3+3*n^2*r^2*s-2*n*p^4-4*n*p^3*s+6*n*p^2*r^2+6*n*p^2*r*s-4*n*p*r^3-2 *n*r^3*s+3*p^4*r+p^4*s-6*p^3*r^2+3*p^2*r^3-3*p^2*r^2*s+2*p*r^3*s-9*n^2*p^2+9*n^ 2*p*r-9*n^2*p*s+9*n^2*r*s+16*n*p^3-6*n*p^2*r+24*n*p^2*s-18*n*p*r^2-30*n*p*r*s+8 *n*r^3+6*n*r^2*s-2*p^4-18*p^3*r-10*p^3*s+33*p^2*r^2+6*p^2*r*s-13*p*r^3+9*p*r^2* s-5*r^3*s+13*n^2*p-6*n^2*r+7*n^2*s-44*n*p^2+18*n*p*r-44*n*p*s+12*n*r^2+30*n*r*s +14*p^3+36*p^2*r+34*p^2*s-57*p*r^2-24*p*r*s+14*r^3-3*r^2*s-6*n^2+50*n*p-12*n*r+ 26*n*s-35*p^2-27*p*r-47*p*s+30*r^2+21*r*s-20*n+37*p+6*r+23*s-14)/(3*n^2*p^2-6*n ^2*p*r+3*n^2*r^2+4*n*p^3-6*n*p^2*r+2*n*r^3+p^4-3*p^2*r^2+2*p*r^3-9*n^2*p+9*n^2* r-12*n*p^2+6*n*p*r+6*n*r^2-2*p^3-6*p^2*r+9*p*r^2-r^3+7*n^2+8*n*p+6*n*r-2*p^2+12 *p*r-3*r^2+2*n+5*p-3*r-1)*diff(A[n](r,s),r)+(n^2*p^4-n^2*p^3*r+3*n^2*p^3*s-3*n^ 2*p^2*r^2-9*n^2*p^2*r*s+5*n^2*p*r^3+9*n^2*p*r^2*s-2*n^2*r^4-3*n^2*r^3*s-4*n*p^4 *r-2*n*p^4*s+10*n*p^3*r^2+2*n*p^3*r*s-6*n*p^2*r^3+6*n*p^2*r^2*s-2*n*p*r^4-10*n* p*r^3*s+2*n*r^5+4*n*r^4*s+3*p^4*r^2+2*p^4*r*s-9*p^3*r^3-5*p^3*r^2*s+9*p^2*r^4+3 *p^2*r^3*s-3*p*r^5+p*r^4*s-r^5*s-7*n^2*p^3+6*n^2*p^2*r-15*n^2*p^2*s+9*n^2*p*r^2 +30*n^2*p*r*s-8*n^2*r^3-15*n^2*r^2*s+2*n*p^4+26*n*p^3*r+20*n*p^3*s-60*n*p^2*r^2 -30*n*p^2*r*s+34*n*p*r^3-2*n*r^4+10*n*r^3*s-4*p^4*r-2*p^4*s-11*p^3*r^2-12*p^3*r *s+42*p^2*r^3+33*p^2*r^2*s-35*p*r^4-22*p*r^3*s+8*r^5+3*r^4*s+17*n^2*p^2-11*n^2* p*r+23*n^2*p*s-6*n^2*r^2-23*n^2*r*s-14*n*p^3-56*n*p^2*r-64*n*p^2*s+108*n*p*r^2+ 82*n*p*r*s-38*n*r^3-18*n*r^2*s+p^4+27*p^3*r+17*p^3*s-6*p^2*r^2+13*p^2*r*s-50*p* r^3-54*p*r^2*s+28*r^4+24*r^3*s-17*n^2*p+6*n^2*r-11*n^2*s+34*n*p^2+46*n*p*r+80*n *p*s-58*n*r^2-58*n*r*s-7*p^3-62*p^2*r-49*p^2*s+48*p*r^2+18*p*r*s+10*r^3+20*r^2* s+6*n^2-34*n*p-12*n*r-34*n*s+17*p^2+57*p*r+57*p*s-34*r^2-23*r*s+12*n-17*p-18*r-\ 23*s+6)/(-3+p-r)/(3*n^2*p^2-6*n^2*p*r+3*n^2*r^2+4*n*p^3-6*n*p^2*r+2*n*r^3+p^4-3 *p^2*r^2+2*p*r^3-9*n^2*p+9*n^2*r-12*n*p^2+6*n*p*r+6*n*r^2-2*p^3-6*p^2*r+9*p*r^2 -r^3+7*n^2+8*n*p+6*n*r-2*p^2+12*p*r-3*r^2+2*n+5*p-3*r-1)*diff(diff(A[n](r,s),r) ,r)-(n^2*p^3*r+n^2*p^3*s-3*n^2*p^2*r^2-3*n^2*p^2*r*s+3*n^2*p*r^3+3*n^2*p*r^2*s- n^2*r^4-n^2*r^3*s-2*n*p^3*r^2-2*n*p^3*r*s+6*n*p^2*r^3+6*n*p^2*r^2*s-6*n*p*r^4-6 *n*p*r^3*s+2*n*r^5+2*n*r^4*s+p^3*r^3+p^3*r^2*s-3*p^2*r^4-3*p^2*r^3*s+3*p*r^5+3* p*r^4*s-r^6-r^5*s-4*n^2*p^2*r-4*n^2*p^2*s+8*n^2*p*r^2+8*n^2*p*r*s-4*n^2*r^3-4*n ^2*r^2*s+2*n*p^3*r+2*n*p^3*s+2*n*p^2*r^2+2*n*p^2*r*s-10*n*p*r^3-10*n*p*r^2*s+6* n*r^4+6*n*r^3*s-2*p^3*r^2-2*p^3*r*s+2*p^2*r^3+2*p^2*r^2*s+2*p*r^4+2*p*r^3*s-2*r ^5-2*r^4*s+5*n^2*p*r+5*n^2*p*s-5*n^2*r^2-5*n^2*r*s-8*n*p^2*r-8*n*p^2*s+6*n*p*r^ 2+6*n*p*r*s+2*n*r^3+2*n*r^2*s+p^3*r+p^3*s+5*p^2*r^2+5*p^2*r*s-8*p*r^3-8*p*r^2*s +2*r^4+2*r^3*s-2*n^2*r-2*n^2*s+10*n*p*r+10*n*p*s-6*n*r^2-6*n*r*s-4*p^2*r-4*p^2* s-2*p*r^2-2*p*r*s+4*r^3+4*r^2*s-4*n*r-4*n*s+5*p*r+5*p*s-r^2-r*s-2*r-2*s)/(3*n^2 *p^3-9*n^2*p^2*r+9*n^2*p*r^2-3*n^2*r^3+4*n*p^4-10*n*p^3*r+6*n*p^2*r^2+2*n*p*r^3 -2*n*r^4+p^5-p^4*r-3*p^3*r^2+5*p^2*r^3-2*p*r^4-18*n^2*p^2+36*n^2*p*r-18*n^2*r^2 -24*n*p^3+36*n*p^2*r-12*n*r^3-5*p^4-4*p^3*r+24*p^2*r^2-16*p*r^3+r^4+34*n^2*p-34 *n^2*r+44*n*p^2-20*n*p*r-24*n*r^2+4*p^3+32*p^2*r-42*p*r^2+6*r^3-21*n^2-22*n*p-\ 20*n*r+11*p^2-44*p*r+12*r^2-6*n-16*p+10*r+3)*diff(diff(diff(A[n](r,s),r),r),r)- (n^2*p^2-2*n^2*p*r+n^2*r^2+2*n*p^3-4*n*p^2*r+2*n*p*r^2+p^4-2*p^3*r+p^2*r^2-4*n^ 2*p+4*n^2*r-8*n*p^2+8*n*p*r-4*p^3+4*p^2*r+4*n^2+8*n*p+4*p^2)/(3*n^2*p^2-6*n^2*p *r+3*n^2*r^2+4*n*p^3-6*n*p^2*r+2*n*r^3+p^4-3*p^2*r^2+2*p*r^3-9*n^2*p+9*n^2*r-12 *n*p^2+6*n*p*r+6*n*r^2-2*p^3-6*p^2*r+9*p*r^2-r^3+7*n^2+8*n*p+6*n*r-2*p^2+12*p*r -3*r^2+2*n+5*p-3*r-1)*A[n+1](r,s)+diff(A[n+1](r,s),r)-(3*n^2*p^3-9*n^2*p^2*r+9* n^2*p*r^2-3*n^2*r^3+2*n*p^4-2*n*p^3*r-6*n*p^2*r^2+10*n*p*r^3-4*n*r^4+2*p^4*r-5* p^3*r^2+3*p^2*r^3+p*r^4-r^5-15*n^2*p^2+30*n^2*p*r-15*n^2*r^2-8*n*p^3-6*n*p^2*r+ 36*n*p*r^2-22*n*r^3+2*p^4-16*p^3*r+21*p^2*r^2-2*p*r^3-5*r^4+23*n^2*p-23*n^2*r+4 *n*p^2+38*n*p*r-42*n*r^2-11*p^3+37*p^2*r-18*p*r^2-8*r^3-11*n^2+12*n*p-34*n*r+19 *p^2-26*p*r-4*r^2-10*n-11*p+r+1)/(3*n^2*p^3-9*n^2*p^2*r+9*n^2*p*r^2-3*n^2*r^3+4 *n*p^4-10*n*p^3*r+6*n*p^2*r^2+2*n*p*r^3-2*n*r^4+p^5-p^4*r-3*p^3*r^2+5*p^2*r^3-2 *p*r^4-18*n^2*p^2+36*n^2*p*r-18*n^2*r^2-24*n*p^3+36*n*p^2*r-12*n*r^3-5*p^4-4*p^ 3*r+24*p^2*r^2-16*p*r^3+r^4+34*n^2*p-34*n^2*r+44*n*p^2-20*n*p*r-24*n*r^2+4*p^3+ 32*p^2*r-42*p*r^2+6*r^3-21*n^2-22*n*p-20*n*r+11*p^2-44*p*r+12*r^2-6*n-16*p+10*r +3)*diff(diff(A[n+1](r,s),r),r)+(n^2*p^3-3*n^2*p^2*r+3*n^2*p*r^2-n^2*r^3+2*n*p^ 3*r-6*n*p^2*r^2+6*n*p*r^3-2*n*r^4+p^3*r^2-3*p^2*r^3+3*p*r^4-r^5-4*n^2*p^2+8*n^2 *p*r-4*n^2*r^2+2*n*p^3-14*n*p^2*r+22*n*p*r^2-10*n*r^3+2*p^3*r-10*p^2*r^2+14*p*r ^3-6*r^4+5*n^2*p-5*n^2*r-8*n*p^2+26*n*p*r-18*n*r^2+p^3-11*p^2*r+24*p*r^2-14*r^3 -2*n^2+10*n*p-14*n*r-4*p^2+18*p*r-16*r^2-4*n+5*p-9*r-2)/(3*n^2*p^3-9*n^2*p^2*r+ 9*n^2*p*r^2-3*n^2*r^3+4*n*p^4-10*n*p^3*r+6*n*p^2*r^2+2*n*p*r^3-2*n*r^4+p^5-p^4* r-3*p^3*r^2+5*p^2*r^3-2*p*r^4-18*n^2*p^2+36*n^2*p*r-18*n^2*r^2-24*n*p^3+36*n*p^ 2*r-12*n*r^3-5*p^4-4*p^3*r+24*p^2*r^2-16*p*r^3+r^4+34*n^2*p-34*n^2*r+44*n*p^2-\ 20*n*p*r-24*n*r^2+4*p^3+32*p^2*r-42*p*r^2+6*r^3-21*n^2-22*n*p-20*n*r+11*p^2-44* p*r+12*r^2-6*n-16*p+10*r+3)*diff(diff(diff(A[n+1](r,s),r),r),r) = 0 (4*n^5+16*n^4*q-12*n^4*s+25*n^3*q^2-36*n^3*q*s+12*n^3*s^2+19*n^2*q^3-39*n^2*q^2 *s+24*n^2*q*s^2-4*n^2*s^3+7*n*q^4-18*n*q^3*s+15*n*q^2*s^2-4*n*q*s^3+q^5-3*q^4*s +3*q^3*s^2-q^2*s^3+8*n^4+26*n^3*q-20*n^3*s+31*n^2*q^2-46*n^2*q*s+16*n^2*s^2+16* n*q^3-34*n*q^2*s+22*n*q*s^2-4*n*s^3+3*q^4-8*q^3*s+7*q^2*s^2-2*q*s^3+n^3+5*n^2*q -7*n^2*s+6*n*q^2-14*n*q*s+7*n*s^2+2*q^3-6*q^2*s+5*q*s^2-s^3-7*n^2-8*n*q+2*n*s-2 *q^2+s^2-5*n-3*q+s-1)/(n-s+1)/(n^3-3*n^2*s-3*n*q^2+3*n*s^2-2*q^3+3*q^2*s-s^3+3* n^2+6*n*q-6*n*s+3*q^2-6*q*s+3*s^2+n+2*q-s-1)*A[n](r,s)-(4*n^5+14*n^4*q-8*n^4*s+ 18*n^3*q^2-16*n^3*q*s+10*n^2*q^3-6*n^2*q^2*s-12*n^2*q*s^2+8*n^2*s^3+2*n*q^4+4*n *q^3*s-18*n*q^2*s^2+16*n*q*s^3-4*n*s^4+2*q^4*s-6*q^3*s^2+6*q^2*s^3-2*q*s^4+11*n ^4+32*n^3*q-20*n^3*s+33*n^2*q^2-36*n^2*q*s+6*n^2*s^2+14*n*q^3-18*n*q^2*s+4*n*s^ 3+2*q^4-2*q^3*s-3*q^2*s^2+4*q*s^3-s^4+8*n^3+20*n^2*q-16*n^2*s+16*n*q^2-24*n*q*s +8*n*s^2+4*q^3-8*q^2*s+4*q*s^2-2*n^2-4*n*s+q^2-4*q*s+2*s^2-4*n-2*q-1)/(n-s+1)/( n^3-3*n^2*s-3*n*q^2+3*n*s^2-2*q^3+3*q^2*s-s^3+3*n^2+6*n*q-6*n*s+3*q^2-6*q*s+3*s ^2+n+2*q-s-1)*diff(A[n](r,s),s)+(n^5+3*n^4*q-n^4*s+3*n^3*q^2-2*n^3*s^2+n^2*q^3+ 3*n^2*q^2*s-6*n^2*q*s^2+2*n^2*s^3+2*n*q^3*s-3*n*q^2*s^2+n*s^4+q^3*s^2-3*q^2*s^3 +3*q*s^4-s^5+4*n^4+10*n^3*q-4*n^3*s+8*n^2*q^2-2*n^2*q*s-4*n^2*s^2+2*n*q^3+4*n*q ^2*s-10*n*q*s^2+4*n*s^3+2*q^3*s-4*q^2*s^2+2*q*s^3+6*n^3+12*n^2*q-6*n^2*s+7*n*q^ 2-4*n*q*s-2*n*s^2+q^3+q^2*s-4*q*s^2+2*s^3+4*n^2+6*n*q-4*n*s+2*q^2-2*q*s+n+q-s)/ (n-s+1)/(n^3-3*n^2*s-3*n*q^2+3*n*s^2-2*q^3+3*q^2*s-s^3+3*n^2+6*n*q-6*n*s+3*q^2-\ 6*q*s+3*s^2+n+2*q-s-1)*diff(diff(A[n](r,s),s),s)+q^2*(n^3+3*n^2*q-3*n^2*s+3*n*q ^2-6*n*q*s+3*n*s^2+q^3-3*q^2*s+3*q*s^2-s^3-n^2-2*n*q+2*n*s-q^2+2*q*s-s^2)/(n+q- s+1)/(n-s+1)/(n^3-3*n^2*s-3*n*q^2+3*n*s^2-2*q^3+3*q^2*s-s^3+3*n^2+6*n*q-6*n*s+3 *q^2-6*q*s+3*s^2+n+2*q-s-1)*A[n+1](r,s)+(2*n^4*q+5*n^3*q^2-8*n^3*q*s+3*n^2*q^3-\ 15*n^2*q^2*s+12*n^2*q*s^2-n*q^4-6*n*q^3*s+15*n*q^2*s^2-8*n*q*s^3-q^5+q^4*s+3*q^ 3*s^2-5*q^2*s^3+2*q*s^4-n^4+4*n^3*s+6*n^2*q^2-6*n^2*s^2+8*n*q^3-12*n*q^2*s+4*n* s^3+3*q^4-8*q^3*s+6*q^2*s^2-s^4-2*n^3-6*n^2*q+6*n^2*s-4*n*q^2+12*n*q*s-6*n*s^2+ 4*q^2*s-6*q*s^2+2*s^3-4*n*q-3*q^2+4*q*s+2*n-2*s+1)/(n+q-s+1)/(n-s+1)/(n^3-3*n^2 *s-3*n*q^2+3*n*s^2-2*q^3+3*q^2*s-s^3+3*n^2+6*n*q-6*n*s+3*q^2-6*q*s+3*s^2+n+2*q- s-1)*diff(A[n+1](r,s),s)+diff(diff(A[n+1](r,s),s),s)-(n^3+2*n^2*q-3*n^2*s+n*q^2 -4*n*q*s+3*n*s^2-q^2*s+2*q*s^2-s^3+2*n^2+3*n*q-4*n*s+q^2-3*q*s+2*s^2+n+q-s)/(n^ 3-3*n^2*s-3*n*q^2+3*n*s^2-2*q^3+3*q^2*s-s^3+3*n^2+6*n*q-6*n*s+3*q^2-6*q*s+3*s^2 +n+2*q-s-1)*diff(diff(diff(A[n+1](r,s),s),s),s) = 0 ------------------------------------------------- This took, 0.140, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ 2 3 (k - 1 + p) A[n](r, s) = ) binomial(n, k) binomial(n + k, k) (r + k) / ----- k = 0 (n - k + q) k (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)^2*binomial(n+k,k)^3*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x ^k,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 3 4 3 3 3 3 3 2 2 3 2 3 3 - (n p - 3 n p r + n p s + 3 n p r - 3 n p r s - n p r 3 2 3 3 2 5 2 4 2 4 2 3 2 + 3 n p r s - n r s - 3 n p + 9 n p r - 3 n p s - 9 n p r 2 3 2 2 3 2 2 2 2 3 6 + 9 n p r s + 3 n p r - 9 n p r s + 3 n p r s + 3 n p 5 5 4 2 4 3 3 3 2 - 9 n p r + 3 n p s + 9 n p r - 9 n p r s - 3 n p r + 9 n p r s 2 3 7 6 6 5 2 5 4 3 - 3 n p r s - p + 3 p r - p s - 3 p r + 3 p r s + p r 4 2 3 3 3 3 3 2 3 2 3 2 - 3 p r s + p r s - 10 n p + 21 n p r - 9 n p s - 12 n p r 3 3 3 3 2 2 4 2 3 2 3 + 18 n p r s + n r - 9 n r s + 36 n p - 81 n p r + 33 n p s 2 2 2 2 2 2 3 2 2 2 3 + 54 n p r - 72 n p r s - 9 n p r + 45 n p r s - 6 n r s 5 4 4 3 2 3 2 3 - 42 n p + 99 n p r - 39 n p s - 72 n p r + 90 n p r s + 15 n p r 2 2 3 6 5 5 4 2 - 63 n p r s + 12 n p r s + 16 p - 39 p r + 15 p s + 30 p r 4 3 3 3 2 2 3 3 2 3 - 36 p r s - 7 p r + 27 p r s - 6 p r s + 35 n p - 44 n p r 3 3 2 3 2 3 2 2 2 2 + 26 n p s + 9 n r - 26 n r s - 165 n p + 258 n p r - 132 n p s 2 2 2 2 3 2 2 4 3 - 99 n p r + 186 n p r s + 6 n r - 54 n r s + 237 n p - 420 n p r 3 2 2 2 3 2 + 198 n p s + 207 n p r - 330 n p r s - 24 n p r + 144 n p r s 3 5 4 4 3 2 3 - 12 n r s - 107 p + 206 p r - 92 p s - 117 p r + 170 p r s 2 3 2 2 3 3 3 3 + 18 p r - 90 p r s + 12 p r s - 50 n p + 26 n r - 24 n s 2 2 2 2 2 2 2 3 + 360 n p - 342 n p r + 228 n p s + 54 n r - 156 n r s - 690 n p 2 2 2 3 + 858 n p r - 492 n p s - 252 n p r + 528 n p r s + 12 n r 2 4 3 3 2 2 2 - 108 n r s + 388 p - 566 p r + 296 p s + 222 p r - 396 p r s 3 2 3 3 2 2 2 - 20 p r + 132 p r s - 8 r s + 24 n - 372 n p + 156 n r - 144 n s 2 2 3 + 1092 n p - 840 n p r + 600 n p s + 108 n r - 312 n r s - 824 p 2 2 2 3 2 2 + 852 p r - 528 p s - 204 p r + 456 p r s + 8 r - 72 r s + 144 n 2 2 - 888 n p + 312 n r - 288 n s + 1024 p - 664 p r + 496 p s + 72 r / - 208 r s + 288 n - 688 p + 208 r - 192 s + 192) A[n](r, s) / ((n + 1) / 3 3 6 3 5 3 5 3 4 2 %2 (p - r - 1) ) + (3 n p - 14 n p r + 4 n p s + 25 n p r 3 4 3 3 3 3 3 2 3 2 4 3 2 3 - 20 n p r s - 20 n p r + 40 n p r s + 5 n p r - 40 n p r s 3 5 3 4 3 6 3 5 2 7 2 6 + 2 n p r + 20 n p r s - n r - 4 n r s - 6 n p + 24 n p r 2 6 2 5 2 2 5 2 4 2 2 3 4 - 9 n p s - 30 n p r + 42 n p r s - 75 n p r s + 30 n p r 2 3 3 2 2 5 2 2 4 2 6 2 5 + 60 n p r s - 24 n p r - 15 n p r s + 6 n p r - 6 n p r s 2 6 8 7 7 6 2 6 + 3 n r s + 3 n p - 6 n p r + 6 n p s - 15 n p r - 24 n p r s 5 3 5 2 4 4 3 5 3 4 + 60 n p r + 30 n p r s - 75 n p r + 42 n p r - 30 n p r s 2 6 2 5 6 8 8 7 2 - 9 n p r + 24 n p r s - 6 n p r s - 4 p r - p s + 20 p r 7 6 3 6 2 5 4 5 3 4 5 + 2 p r s - 40 p r + 5 p r s + 40 p r - 20 p r s - 20 p r 4 4 3 6 3 5 2 6 3 5 3 4 + 25 p r s + 4 p r - 14 p r s + 3 p r s - 39 n p + 149 n p r 3 4 3 3 2 3 3 3 2 3 - 46 n p s - 206 n p r + 184 n p r s + 114 n p r 3 2 2 3 4 3 3 3 5 3 4 - 276 n p r s - 11 n p r + 184 n p r s - 7 n r - 46 n r s 2 6 2 5 2 5 2 4 2 2 4 + 96 n p - 330 n p r + 129 n p s + 348 n p r - 507 n p r s 2 3 3 2 3 2 2 2 4 2 2 3 - 12 n p r + 738 n p r s - 192 n p r - 462 n p r s 2 5 2 4 2 6 2 5 7 6 + 102 n p r + 93 n p r s - 12 n r + 9 n r s - 60 n p + 123 n p r 6 5 2 5 4 3 4 2 - 105 n p s + 147 n p r + 372 n p r s - 618 n p r - 423 n p r s 3 4 3 3 2 5 2 4 6 + 642 n p r + 72 n p r s - 273 n p r + 177 n p r s + 39 n p r 5 6 8 7 7 6 2 - 108 n p r s + 15 n r s + 3 p + 58 p r + 22 p s - 289 p r 6 5 3 5 2 4 4 4 3 3 5 - 49 p r s + 516 p r - 39 p r s - 439 p r + 206 p r s + 178 p r 3 4 2 6 2 5 6 3 4 - 224 p r s - 27 p r + 99 p r s - 15 p r s + 202 n p 3 3 3 3 3 2 2 3 2 3 3 - 606 n p r + 202 n p s + 606 n p r - 606 n p r s - 202 n p r 3 2 3 3 2 5 2 4 2 4 + 606 n p r s - 202 n r s - 636 n p + 1830 n p r - 744 n p s 2 3 2 2 3 2 2 3 2 2 2 - 1566 n p r + 2370 n p r s + 78 n p r - 2646 n p r s 2 4 2 3 2 5 2 4 6 + 402 n p r + 1158 n p r s - 108 n r - 138 n r s + 507 n p 5 5 4 2 4 3 3 - 1005 n p r + 765 n p s - 486 n p r - 2337 n p r s + 2460 n p r 3 2 2 4 2 3 5 + 2304 n p r s - 2019 n p r - 540 n p r s + 585 n p r 4 6 5 7 6 6 - 309 n p r s - 42 n r + 117 n r s - 54 p - 333 p r - 204 p s 5 2 5 4 3 4 2 3 4 + 1731 p r + 459 p r s - 2716 p r + 21 p r s + 1902 p r 3 3 2 5 2 4 6 5 6 - 796 p r s - 591 p r + 732 p r s + 61 p r - 231 p r s + 19 r s 3 3 3 2 3 2 3 2 3 - 535 n p + 1178 n p r - 427 n p s - 751 n p r + 854 n p r s 3 3 3 2 2 4 2 3 2 3 + 108 n r - 427 n r s + 2259 n p - 5220 n p r + 2211 n p s 2 2 2 2 2 2 3 2 2 + 3387 n p r - 5352 n p r s - 150 n p r + 4071 n p r s 2 4 2 3 5 4 4 - 276 n r - 930 n r s - 2361 n p + 4284 n p r - 3003 n p s 3 2 3 2 3 2 2 + 447 n p r + 7590 n p r s - 4716 n p r - 6033 n p r s 4 3 5 4 6 5 + 2760 n p r + 1308 n p r s - 414 n r + 138 n r s + 414 p + 919 p r 5 4 2 4 3 3 3 2 + 1042 p s - 5543 p r - 2207 p r s + 7448 p r + 619 p r s 2 4 2 3 5 4 6 5 - 4059 p r + 1392 p r s + 867 p r - 1023 p r s - 46 r + 177 r s 3 2 3 3 3 2 3 + 767 n p - 1093 n p r + 441 n p s + 326 n r - 441 n r s 2 3 2 2 2 2 2 2 2 - 4644 n p + 8049 n p r - 3582 n p s - 3489 n p r + 5841 n p r s 2 3 2 2 4 3 3 + 84 n r - 2259 n r s + 6621 n p - 10341 n p r + 6855 n p s 2 2 2 3 2 4 + 762 n p r - 13401 n p r s + 4338 n p r + 7560 n p r s - 1380 n r 3 5 4 4 3 2 - 1014 n r s - 1764 p - 1009 p r - 3208 p s + 10177 p r 3 2 3 2 2 4 3 + 5977 p r s - 11186 p r - 2265 p r s + 4256 p r - 1010 p r s 5 4 3 3 3 2 2 - 474 r + 506 r s - 566 n p + 386 n r - 180 n s + 5529 n p 2 2 2 2 2 3 - 6339 n p r + 3021 n p s + 1350 n r - 2481 n r s - 11445 n p 2 2 2 3 + 14166 n p r - 9111 n p s - 1737 n p r + 12180 n p r s - 1524 n r 2 4 3 3 2 2 2 - 3609 n r s + 4564 p - 711 p r + 6100 p s - 10611 p r - 9189 p r s 3 2 4 3 3 2 + 8686 p r + 3099 p r s - 1748 r + 170 r s + 168 n - 3534 n p 2 2 2 2 + 1986 n r - 1044 n s + 11913 n p - 10203 n p r + 6555 n p s + 882 n r 3 2 2 2 - 4467 n r s - 7336 p + 3087 p r - 7008 p s + 5745 p r + 7461 p r s 3 2 2 2 - 2708 r - 1497 r s + 936 n - 6834 n p + 2982 n r - 1980 n s + 7151 p 2 - 3005 p r + 4463 p s - 1230 r - 2483 r s + 1656 n - 3866 p + 998 r /d \ / - 1212 s + 888) |-- A[n](r, s)| / ((n p - n r - n + p - r - 1) \dr / / 2 2 2 3 6 3 5 3 5 (p - 2 p r + r - 3 p + 3 r + 2) %2) - (3 n p - 12 n p r + 6 n p s 3 4 2 3 4 3 3 2 3 2 4 3 2 3 + 15 n p r - 30 n p r s + 60 n p r s - 15 n p r - 60 n p r s 3 5 3 4 3 6 3 5 2 7 2 6 + 12 n p r + 30 n p r s - 3 n r - 6 n r s - 3 n p + 3 n p r 2 6 2 5 2 2 5 2 4 3 2 4 2 - 9 n p s + 27 n p r + 36 n p r s - 75 n p r - 45 n p r s 2 3 4 2 2 5 2 2 4 2 6 2 5 + 75 n p r - 27 n p r + 45 n p r s - 3 n p r - 36 n p r s 2 7 2 6 7 7 6 2 6 + 3 n r + 9 n r s + 9 n p r + 3 n p s - 36 n p r - 3 n p r s 5 3 5 2 4 3 3 5 3 4 + 45 n p r - 27 n p r s + 75 n p r s - 45 n p r - 75 n p r s 2 6 2 5 7 6 7 7 2 + 36 n p r + 27 n p r s - 9 n p r + 3 n p r s - 3 n r s - 6 p r 7 6 3 6 2 5 4 5 3 4 5 - 3 p r s + 30 p r + 12 p r s - 60 p r - 15 p r s + 60 p r 3 6 3 5 2 7 2 6 7 3 5 - 30 p r + 15 p r s + 6 p r - 12 p r s + 3 p r s - 42 n p 3 4 3 4 3 3 2 3 3 3 2 3 + 138 n p r - 72 n p s - 132 n p r + 288 n p r s - 12 n p r 3 2 2 3 4 3 3 3 5 3 4 - 432 n p r s + 78 n p r + 288 n p r s - 30 n r - 72 n r s 2 6 2 5 2 5 2 4 2 2 4 + 57 n p - 72 n p r + 144 n p s - 279 n p r - 504 n p r s 2 3 3 2 3 2 2 2 4 2 2 3 + 696 n p r + 576 n p r s - 549 n p r - 144 n p r s 2 5 2 4 2 6 2 5 7 6 + 144 n p r - 144 n p r s + 3 n r + 72 n r s - 6 n p - 138 n p r 6 5 2 5 4 3 4 2 - 66 n p s + 540 n p r + 108 n p r s - 636 n p r + 234 n p r s 3 4 3 3 2 5 2 4 6 + 114 n p r - 696 n p r s + 270 n p r + 594 n p r s - 168 n p r 5 7 6 7 7 6 2 - 180 n p r s + 24 n r + 6 n r s + 9 p r + 3 p s + 60 p r 6 5 3 5 2 4 4 4 3 + 45 p r s - 363 p r - 189 p r s + 672 p r + 237 p r s 3 5 3 4 2 6 2 5 7 6 - 573 p r - 63 p r s + 228 p r - 81 p r s - 33 p r + 57 p r s 7 3 4 3 3 3 3 3 2 2 - 9 r s + 236 n p - 611 n p r + 333 n p s + 417 n p r 3 2 3 3 3 2 3 4 3 3 - 999 n p r s + 55 n p r + 999 n p r s - 97 n r - 333 n r s 2 5 2 4 2 4 2 3 2 2 3 - 444 n p + 588 n p r - 924 n p s + 1089 n p r + 2697 n p r s 2 2 3 2 2 2 2 4 2 3 - 2355 n p r - 2547 n p r s + 1311 n p r + 699 n p r s 2 5 2 4 6 5 5 - 189 n r + 75 n r s + 105 n p + 846 n p r + 588 n p s 4 2 4 3 3 3 2 - 3249 n p r - 1092 n p r s + 3435 n p r - 513 n p r s 2 4 2 3 5 4 6 - 846 n p r + 2211 n p r s - 477 n p r - 1455 n p r s + 186 n r 5 7 6 6 5 2 5 + 261 n r s - 3 p - 141 p r - 57 p s - 123 p r - 246 p r s 4 3 4 2 3 4 3 3 2 5 + 1675 p r + 1161 p r s - 2878 p r - 1377 p r s + 1992 p r 2 4 6 5 7 6 3 3 + 480 p r s - 568 p r + 99 p r s + 46 r - 60 r s - 680 n p 3 2 3 2 3 2 3 3 3 + 1296 n p r - 744 n p s - 552 n p r + 1488 n p r s - 64 n r 3 2 2 4 2 3 2 3 2 2 2 - 744 n r s + 1836 n p - 2265 n p r + 3039 n p s - 1989 n p r 2 2 2 3 2 2 2 4 - 6885 n p r s + 3429 n p r + 4653 n p r s - 1011 n r 2 3 5 4 4 3 2 - 807 n r s - 762 n p - 2622 n p r - 2760 n p s + 9990 n p r 3 2 3 2 2 4 + 4962 n p r s - 8850 n p r - 558 n p r s + 2028 n p r 3 5 4 6 5 5 - 2730 n p r s + 216 n r + 1086 n r s + 51 p + 906 p r + 450 p s 4 2 4 3 3 3 2 2 4 - 699 p r + 510 p r s - 3565 p r - 3501 p r s + 5808 p r 2 3 5 4 6 5 3 2 + 3687 p r s - 2961 p r - 1161 p r s + 460 r + 15 r s + 1057 n p 3 3 3 2 3 2 3 - 1309 n p r + 805 n p s + 252 n r - 805 n r s - 4347 n p 2 2 2 2 2 2 2 2 3 + 4467 n p r - 5403 n p s + 1692 n p r + 8391 n p r s - 1812 n r 2 2 4 3 3 2 2 - 2988 n r s + 2964 n p + 4224 n p r + 7386 n p s - 16479 n p r 2 3 2 4 3 - 11352 n p r s + 10845 n p r + 2961 n p r s - 1554 n r + 1005 n r s 5 4 4 3 2 3 2 3 - 360 p - 3072 p r - 1908 p s + 4155 p r + 246 p r s + 3107 p r 2 2 4 3 5 4 3 + 5307 p r s - 5396 p r - 4525 p r s + 1566 r + 880 r s - 838 n p 3 3 2 2 2 2 2 2 + 498 n r - 340 n s + 5883 n p - 4323 n p r + 4929 n p s - 540 n r 2 3 2 2 2 - 3909 n r s - 6654 n p - 3090 n p r - 11286 n p s + 13770 n p r 3 2 4 3 + 12714 n p r s - 5046 n r - 2448 n r s + 1364 p + 5878 p r 3 2 2 2 3 2 4 + 4680 p s - 8649 p r - 2754 p r s - 25 p r - 3603 p r s + 1772 r 3 3 2 2 2 2 + 2017 r s + 264 n - 4206 n p + 1602 n r - 1812 n s + 8595 n p 2 3 2 + 357 n p r + 9135 n p s - 4536 n r - 5511 n r s - 2987 p - 6261 p r 2 2 3 2 2 - 6627 p s + 8142 p r + 4119 p r s - 970 r + 696 r s + 1224 n 2 2 - 5898 n p + 414 n r - 3036 n s + 3769 p + 3371 p r + 5011 p s - 2880 r / 2 \ |d | / - 1975 r s + 1656 n - 2530 p - 690 r - 1564 s + 696) |--- A[n](r, s)| / | 2 | / \dr / 2 2 2 2 ((n p - 2 n p r + n r - 3 n p + 3 n r + p - 2 p r + r + 2 n - 3 p + 3 r 3 4 3 3 3 2 2 3 2 3 3 + 2) %1 %2) + (n p + 4 n p s - 6 n p r - 12 n p r s + 8 n p r 3 2 3 4 3 3 2 4 2 4 2 3 2 + 12 n p r s - 3 n r - 4 n r s - 6 n p r - 3 n p s + 12 n p r 2 2 2 2 4 2 3 2 5 2 4 + 18 n p r s - 12 n p r - 24 n p r s + 6 n r + 9 n r s 4 2 4 3 3 3 2 2 4 + 9 n p r + 6 n p r s - 24 n p r - 12 n p r s + 18 n p r 4 6 5 4 3 4 2 3 4 + 12 n p r s - 3 n r - 6 n r s - 4 p r - 3 p r s + 12 p r 3 3 2 5 2 4 6 6 3 3 3 2 + 8 p r s - 12 p r - 6 p r s + 4 p r + r s - 10 n p - 30 n p s 3 2 3 3 3 3 2 2 4 2 3 + 30 n p r + 60 n p r s - 20 n r - 30 n r s + 3 n p + 60 n p r 2 3 2 2 2 2 2 2 3 2 2 + 42 n p s - 108 n p r - 36 n p r s + 24 n p r - 54 n p r s 2 4 2 3 4 4 3 2 3 + 21 n r + 48 n r s - 12 n p r - 6 n p s - 66 n p r - 60 n p r s 2 3 2 2 4 3 5 + 180 n p r + 126 n p r s - 114 n p r - 48 n p r s + 12 n r 4 4 2 4 3 3 3 2 2 4 - 12 n r s + 9 p r + 6 p r s + 16 p r + 18 p r s - 72 p r 2 3 5 4 6 5 3 2 3 - 60 p r s + 60 p r + 42 p r s - 13 r - 6 r s + 35 n p + 70 n p s 3 2 3 2 3 2 2 2 2 - 35 n r - 70 n r s - 30 n p - 210 n p r - 195 n p s 2 2 2 2 3 2 2 4 3 + 300 n p r + 180 n p r s - 60 n r + 15 n r s + 3 n p + 120 n p r 3 2 2 2 3 2 + 72 n p s + 117 n p r + 174 n p r s - 396 n p r - 354 n p r s 4 3 4 4 3 2 3 + 156 n r + 108 n r s - 6 p r - 3 p s - 78 p r - 60 p r s 2 3 2 2 4 3 5 4 3 + 40 p r + 3 p r s + 108 p r + 116 p r s - 64 r - 56 r s - 50 n p 3 2 2 2 2 2 2 2 - 50 n s + 105 n p + 300 n p r + 360 n p s - 255 n r - 210 n r s 3 2 2 2 3 - 30 n p - 420 n p r - 300 n p s + 60 n p r - 120 n p r s + 240 n r 2 4 3 3 2 2 2 3 + 270 n r s + p + 60 p r + 34 p s + 219 p r + 198 p r s - 212 p r 2 4 3 3 2 2 2 - 138 p r s - 18 r - 44 r s + 24 n - 150 n p - 144 n r - 222 n s 2 2 3 + 105 n p + 600 n p r + 510 n p s - 189 n r - 66 n r s - 10 p 2 2 2 3 2 2 - 210 p r - 135 p s - 210 p r - 240 p r s + 184 r + 153 r s + 72 n 2 2 - 150 n p - 288 n r - 294 n s + 35 p + 300 p r + 220 p s + 31 r + 74 r s / 3 \ |d | / 3 2 + 72 n - 50 p - 144 r - 122 s + 24) |--- A[n](r, s)| / ((n p - 3 n p r | 3 | / \dr / 2 3 2 2 3 2 2 3 + 3 n p r - n r - 6 n p + 12 n p r - 6 n r + p - 3 p r + 3 p r - r 2 2 + 11 n p - 11 n r - 6 p + 12 p r - 6 r - 6 n + 11 p - 11 r - 6) 2 3 3 2 2 2 3 2 4 (n + r + 1) ) - (n r + n s - 3 n r - 3 n r s + 3 n r + 3 n r s - r 3 2 2 2 3 2 - r s + 3 n r + 3 n s - 6 n r - 6 n r s + 3 r + 3 r s + 3 n r / 4 \ 2 |d | / + 3 n s - 3 r - 3 r s + r + s) |--- A[n](r, s)| / ( | 4 | / \dr / 3 2 2 2 2 2 3 n + 2 n r + n r + 3 n + 4 n r + r + 3 n + 2 r + 1) + (n p 2 2 2 2 2 3 4 3 2 2 3 - 3 n p r + 3 n p r - n r + 2 n p - 6 n p r + 6 n p r - 2 n p r 5 4 3 2 2 3 2 2 2 2 2 3 + p - 3 p r + 3 p r - p r - 9 n p + 18 n p r - 9 n r - 18 n p 2 2 4 3 2 2 2 2 + 36 n p r - 18 n p r - 9 p + 18 p r - 9 p r + 26 n p - 26 n r 2 3 2 2 2 + 52 n p - 52 n p r + 26 p - 26 p r - 24 n - 48 n p - 24 p ) / 3 2 5 2 4 2 3 2 A[n + 1](r, s) / (%2 (p - r - 1) ) - (4 n p - 20 n p r + 40 n p r / 2 2 3 2 4 2 5 6 5 4 2 - 40 n p r + 20 n p r - 4 n r + 6 n p - 28 n p r + 50 n p r 3 3 2 4 5 6 7 6 5 2 - 40 n p r + 10 n p r + 4 n p r - 2 n r + 2 p - 8 p r + 10 p r 3 4 2 5 6 2 4 2 3 2 2 2 - 10 p r + 8 p r - 2 p r - 46 n p + 184 n p r - 276 n p r 2 3 2 4 5 4 3 2 + 184 n p r - 46 n r - 70 n p + 258 n p r - 332 n p r 2 3 4 5 6 5 4 2 + 148 n p r + 18 n p r - 22 n r - 23 p + 68 p r - 41 p r 3 3 2 4 5 6 2 3 2 2 - 56 p r + 79 p r - 28 p r + r + 202 n p - 606 n p r 2 2 2 3 4 3 2 2 + 606 n p r - 202 n r + 312 n p - 844 n p r + 660 n p r 3 4 5 4 3 2 2 3 4 - 36 n p r - 92 n r + 99 p - 183 p r - 56 p r + 276 p r - 147 p r 5 2 2 2 2 2 3 2 + 11 r - 427 n p + 854 n p r - 427 n r - 666 n p + 1144 n p r 2 3 4 3 2 2 3 4 - 290 n p r - 188 n r - 193 p + 106 p r + 413 p r - 372 p r + 46 r 2 2 2 2 3 + 441 n p - 441 n r + 680 n p - 478 n p r - 202 n r + 145 p 2 2 3 2 2 + 245 p r - 484 p r + 94 r - 180 n - 250 n p - 110 n r + 31 p 2 /d \ / - 312 p r + 101 r - 24 n - 79 p + 55 r + 12) |-- A[n + 1](r, s)| / ( \dr / / 2 2 2 2 2 5 (p - r - 1) (n + r + 1) (p - 2 p r + r - 3 p + 3 r + 2) ) + (6 n p 2 4 2 3 2 2 2 3 2 4 2 5 6 - 30 n p r + 60 n p r - 60 n p r + 30 n p r - 6 n r + 6 n p 5 4 2 2 4 5 6 7 6 - 24 n p r + 30 n p r - 30 n p r + 24 n p r - 6 n r + p - p r 5 2 4 3 3 4 2 5 6 7 2 4 - 9 p r + 25 p r - 25 p r + 9 p r + p r - r - 72 n p 2 3 2 2 2 2 3 2 4 5 + 288 n p r - 432 n p r + 288 n p r - 72 n r - 72 n p 4 3 2 2 3 4 5 6 + 216 n p r - 144 n p r - 144 n p r + 216 n p r - 72 n r - 10 p 5 4 2 3 3 2 4 5 6 - 12 p r + 138 p r - 232 p r + 138 p r - 12 p r - 10 r 2 3 2 2 2 2 2 3 4 + 333 n p - 999 n p r + 999 n p r - 333 n r + 328 n p 3 2 2 3 4 5 4 - 646 n p r - 30 n p r + 686 n p r - 338 n r + 28 p + 188 p r 3 2 2 3 4 5 2 2 2 - 699 p r + 689 p r - 173 p r - 33 r - 744 n p + 1488 n p r 2 2 3 2 2 3 4 - 744 n r - 694 n p + 594 n p r + 894 n p r - 794 n r + 24 p 3 2 2 3 4 2 2 - 790 p r + 1482 p r - 690 p r - 26 r + 805 n p - 805 n r 2 2 3 2 2 3 + 626 n p + 358 n p r - 984 n r - 258 p + 1400 p r - 1221 p r + 79 r 2 2 2 - 340 n - 66 n p - 614 n r + 466 p - 998 p r + 192 r - 152 n - 307 p / 2 \ |d | / 2 2 + 155 r + 44) |--- A[n + 1](r, s)| / ((p - 2 p r + r - 3 p + 3 r + 2) | 2 | / \dr / 3 2 2 3 4 3 3 %1 %2) - 2 (2 n p - 6 n p r + 6 n p r - 2 n r + p - 2 p r + 2 p r 4 2 2 3 2 2 3 - r - 15 n p + 30 n p r - 15 n r - 8 p + 9 p r + 6 p r - 7 r 2 2 + 35 n p - 35 n r + 20 p - 5 p r - 15 r - 25 n - 15 p - 10 r - 1) / 3 \ / 4 \ |d | |d | |--- A[n + 1](r, s)|/(%1 (n + r + 1)) + |--- A[n + 1](r, s)| = 0 | 3 | | 4 | \dr / \dr / 3 2 2 3 2 2 %1 := p - 3 p r + 3 p r - r - 6 p + 12 p r - 6 r + 11 p - 11 r - 6 2 2 %2 := n + 2 n r + r + 2 n + 2 r + 1 7 6 6 5 2 5 5 2 4 3 (8 n + 44 n q - 32 n s + 102 n q - 144 n q s + 48 n s + 129 n q 4 2 4 2 4 3 3 4 3 3 - 264 n q s + 168 n q s - 32 n s + 96 n q - 252 n q s 3 2 2 3 3 3 4 2 5 2 4 + 228 n q s - 80 n q s + 8 n s + 42 n q - 132 n q s 2 3 2 2 2 3 2 4 6 5 + 150 n q s - 72 n q s + 12 n q s + 10 n q - 36 n q s 4 2 3 3 2 4 7 6 5 2 4 3 + 48 n q s - 28 n q s + 6 n q s + q - 4 q s + 6 q s - 4 q s 3 4 6 5 5 4 2 4 4 2 + q s + 44 n + 204 n q - 144 n s + 387 n q - 528 n q s + 168 n s 3 3 3 2 3 2 3 3 2 4 + 384 n q - 756 n q s + 456 n q s - 80 n s + 210 n q 2 3 2 2 2 2 3 2 4 5 - 528 n q s + 450 n q s - 144 n q s + 12 n s + 60 n q 4 3 2 2 3 4 6 5 - 180 n q s + 192 n q s - 84 n q s + 12 n q s + 7 q - 24 q s 4 2 3 3 2 4 5 4 4 3 2 + 30 q s - 16 q s + 3 q s + 102 n + 387 n q - 264 n s + 576 n q 3 3 2 2 3 2 2 2 2 - 756 n q s + 228 n s + 420 n q - 792 n q s + 450 n q s 2 3 4 3 2 2 3 4 - 72 n s + 150 n q - 360 n q s + 288 n q s - 84 n q s + 6 n s 5 4 3 2 2 3 4 4 3 + 21 q - 60 q s + 60 q s - 24 q s + 3 q s + 129 n + 384 n q 3 2 2 2 2 2 3 2 - 252 n s + 420 n q - 528 n q s + 150 n s + 200 n q - 360 n q s 2 3 4 3 2 2 3 4 3 + 192 n q s - 28 n s + 35 q - 80 q s + 60 q s - 16 q s + s + 96 n 2 2 2 2 3 2 + 210 n q - 132 n s + 150 n q - 180 n q s + 48 n s + 35 q - 60 q s 2 3 2 2 2 + 30 q s - 4 s + 42 n + 60 n q - 36 n s + 21 q - 24 q s + 6 s + 10 n 9 8 8 + 7 q - 4 s + 1) A[n](r, s)/((n + 1) %1) - (12 n + 84 n q - 60 n s 7 2 7 7 2 6 3 6 2 + 255 n q - 348 n q s + 108 n s + 438 n q - 843 n q s 6 2 6 3 5 4 5 3 5 2 2 + 468 n q s - 60 n s + 465 n q - 1092 n q s + 747 n q s 5 3 5 4 4 5 4 4 4 3 2 - 60 n q s - 60 n s + 312 n q - 795 n q s + 450 n q s 4 2 3 4 4 4 5 3 6 3 5 + 345 n q s - 420 n q s + 108 n s + 129 n q - 300 n q s 3 4 2 3 3 3 3 2 4 3 5 3 6 - 90 n q s + 840 n q s - 915 n q s + 396 n q s - 60 n s 2 7 2 6 2 5 2 2 4 3 2 3 4 + 30 n q - 33 n q s - 252 n q s + 750 n q s - 870 n q s 2 2 5 2 6 2 7 8 7 6 2 + 495 n q s - 132 n q s + 12 n s + 3 n q + 12 n q s - 117 n q s 5 3 4 4 3 5 2 6 7 + 300 n q s - 375 n q s + 252 n q s - 87 n q s + 12 n q s 8 7 2 6 3 5 4 4 5 3 6 2 7 + 3 q s - 18 q s + 45 q s - 60 q s + 45 q s - 18 q s + 3 q s 8 7 7 6 2 6 6 2 + 54 n + 339 n q - 246 n s + 912 n q - 1275 n q s + 414 n s 5 3 5 2 5 2 5 3 4 4 + 1368 n q - 2736 n q s + 1647 n q s - 270 n s + 1245 n q 4 3 4 2 2 4 3 4 4 3 5 - 3120 n q s + 2520 n q s - 615 n q s - 30 n s + 699 n q 3 4 3 3 2 3 2 3 3 4 3 5 - 2010 n q s + 1800 n q s - 240 n q s - 375 n q s + 126 n s 2 6 2 5 2 4 2 2 3 3 2 2 4 + 234 n q - 711 n q s + 540 n q s + 360 n q s - 720 n q s 2 5 2 6 7 6 5 2 4 3 + 351 n q s - 54 n s + 42 n q - 120 n q s + 9 n q s + 330 n q s 3 4 2 5 6 7 8 7 - 480 n q s + 288 n q s - 75 n q s + 6 n s + 3 q - 6 q s 6 2 5 3 4 4 3 5 2 6 7 7 - 18 q s + 75 q s - 105 q s + 72 q s - 24 q s + 3 q s + 85 n 6 6 5 2 5 5 2 4 3 + 478 n q - 361 n s + 1134 n q - 1668 n q s + 585 n s + 1468 n q 4 2 4 2 4 3 3 4 3 3 - 3138 n q s + 2106 n q s - 429 n s + 1117 n q - 3064 n q s 3 2 2 3 3 3 4 2 5 2 4 + 2916 n q s - 1080 n q s + 111 n s + 498 n q - 1629 n q s 2 3 2 2 2 3 2 4 2 5 6 + 1920 n q s - 924 n q s + 114 n q s + 21 n s + 120 n q 5 4 2 3 3 2 4 5 - 444 n q s + 591 n q s - 296 n q s - 18 n q s + 60 n q s 6 7 6 5 2 4 3 3 4 2 5 - 13 n s + 12 q - 48 q s + 66 q s - 23 q s - 28 q s + 30 q s 6 7 6 5 5 4 2 4 - 10 q s + s + 37 n + 201 n q - 180 n s + 447 n q - 783 n q s 4 2 3 3 3 2 3 2 3 3 + 333 n s + 520 n q - 1332 n q s + 1098 n q s - 288 n s 2 4 2 3 2 2 2 2 3 2 4 + 333 n q - 1104 n q s + 1314 n q s - 654 n q s + 111 n s 5 4 3 2 2 3 4 5 + 111 n q - 444 n q s + 672 n q s - 468 n q s + 141 n q s - 12 n s 6 5 4 2 3 3 2 4 5 6 5 + 15 q - 69 q s + 123 q s - 104 q s + 39 q s - 3 q s - s - 39 n 4 4 3 2 3 3 2 2 3 - 132 n q + 69 n s - 165 n q + 132 n q s + 6 n s - 90 n q 2 2 2 2 2 3 4 3 2 2 + 45 n q s + 108 n q s - 66 n s - 18 n q - 36 n q s + 153 n q s 3 4 4 3 2 2 3 4 5 - 132 n q s + 33 n s - 18 q s + 54 q s - 57 q s + 24 q s - 3 s 4 3 3 2 2 2 2 2 - 43 n - 139 n q + 106 n s - 162 n q + 231 n q s - 72 n s 3 2 2 3 4 3 2 2 - 80 n q + 156 n q s - 81 n q s + 6 n s - 14 q + 32 q s - 18 q s 3 4 3 2 2 2 2 - 3 q s + 3 s - n - 18 n q + 33 n s - 24 n q + 60 n q s - 27 n s 3 2 2 3 2 2 2 - 8 q + 24 q s - 18 q s + 3 s + 15 n + 15 n q + 3 q + 3 q s - 3 s /d \ / 5 + 7 n + 4 q - s + 1) |-- A[n](r, s)| / ((n + 1) (-s - 1 + n + q) (2 n q \ds / / 4 2 4 3 3 3 2 3 2 2 4 + 6 n q - 10 n q s + 4 n q - 24 n q s + 20 n q s - 4 n q 2 3 2 2 2 2 3 5 4 3 2 - 12 n q s + 36 n q s - 20 n q s - 6 n q + 8 n q s + 12 n q s 2 3 4 6 5 4 2 3 3 2 4 - 24 n q s + 10 n q s - 2 q + 6 q s - 4 q s - 4 q s + 6 q s 5 5 4 4 3 2 3 3 2 2 3 - 2 q s - n + n q + 5 n s + 12 n q - 4 n q s - 10 n s + 20 n q 2 2 2 2 2 3 4 3 2 2 - 36 n q s + 6 n q s + 10 n s + 13 n q - 40 n q s + 36 n q s 3 4 5 4 3 2 2 3 4 5 - 4 n q s - 5 n s + 3 q - 13 q s + 20 q s - 12 q s + q s + s 4 3 3 2 2 2 2 2 3 - 3 n - 8 n q + 12 n s - 4 n q + 24 n q s - 18 n s + 4 n q 2 2 3 4 3 2 2 3 4 + 8 n q s - 24 n q s + 12 n s + 3 q - 4 q s - 4 q s + 8 q s - 3 s 3 2 2 2 2 3 2 - 2 n - 10 n q + 6 n s - 11 n q + 20 n q s - 6 n s - 3 q + 11 q s 2 3 2 2 2 - 10 q s + 2 s + 2 n - 2 n q - 4 n s - 3 q + 2 q s + 2 s + 3 n + q 10 9 9 8 2 8 - 3 s + 1)) + (6 n + 45 n q - 30 n s + 147 n q - 183 n q s 8 2 7 3 7 2 7 2 6 4 + 48 n s + 273 n q - 462 n q s + 192 n q s + 315 n q 6 3 6 2 2 6 3 6 4 5 5 - 609 n q s + 210 n q s + 168 n q s - 84 n s + 231 n q 5 4 5 3 2 5 2 3 5 4 5 5 - 420 n q s - 147 n q s + 714 n q s - 462 n q s + 84 n s 4 6 4 5 4 4 2 4 3 3 4 2 4 + 105 n q - 105 n q s - 525 n q s + 1155 n q s - 840 n q s 4 5 3 7 3 6 3 5 2 3 4 3 + 210 n q s + 27 n q + 42 n q s - 462 n q s + 840 n q s 3 3 4 3 2 5 3 6 3 7 2 8 - 525 n q s - 42 n q s + 168 n q s - 48 n s + 3 n q 2 7 2 6 2 2 5 3 2 4 4 2 3 5 + 33 n q s - 168 n q s + 210 n q s + 105 n q s - 483 n q s 2 2 6 2 7 2 8 8 7 2 + 462 n q s - 192 n q s + 30 n s + 6 n q s - 15 n q s 6 3 5 4 4 5 3 6 2 7 - 42 n q s + 231 n q s - 420 n q s + 399 n q s - 210 n q s 8 9 8 2 7 3 6 4 5 5 + 57 n q s - 6 n s + 3 q s - 21 q s + 63 q s - 105 q s 4 6 3 7 2 8 9 9 8 8 + 105 q s - 63 q s + 21 q s - 3 q s + 24 n + 168 n q - 120 n s 7 2 7 7 2 6 3 6 2 + 510 n q - 696 n q s + 216 n s + 876 n q - 1686 n q s 6 2 6 3 5 4 5 3 5 2 2 + 936 n q s - 120 n s + 930 n q - 2184 n q s + 1494 n q s 5 3 5 4 4 5 4 4 4 3 2 - 120 n q s - 120 n s + 624 n q - 1590 n q s + 900 n q s 4 2 3 4 4 4 5 3 6 3 5 + 690 n q s - 840 n q s + 216 n s + 258 n q - 600 n q s 3 4 2 3 3 3 3 2 4 3 5 3 6 - 180 n q s + 1680 n q s - 1830 n q s + 792 n q s - 120 n s 2 7 2 6 2 5 2 2 4 3 2 3 4 + 60 n q - 66 n q s - 504 n q s + 1500 n q s - 1740 n q s 2 2 5 2 6 2 7 8 7 6 2 + 990 n q s - 264 n q s + 24 n s + 6 n q + 24 n q s - 234 n q s 5 3 4 4 3 5 2 6 7 + 600 n q s - 750 n q s + 504 n q s - 174 n q s + 24 n q s 8 7 2 6 3 5 4 4 5 3 6 2 7 + 6 q s - 36 q s + 90 q s - 120 q s + 90 q s - 36 q s + 6 q s 8 7 7 6 2 6 6 2 + 20 n + 145 n q - 130 n s + 449 n q - 781 n q s + 326 n s 5 3 5 2 5 2 5 3 4 4 + 773 n q - 1944 n q s + 1545 n q s - 378 n s + 805 n q 4 3 4 2 2 4 3 4 4 3 5 - 2575 n q s + 2865 n q s - 1245 n q s + 150 n s + 515 n q 3 4 3 3 2 3 2 3 3 4 3 5 - 1930 n q s + 2570 n q s - 1320 n q s + 75 n q s + 90 n s 2 6 2 5 2 4 2 2 3 3 2 2 4 + 195 n q - 795 n q s + 1080 n q s - 310 n q s - 525 n q s 2 5 2 6 7 6 5 2 + 465 n q s - 110 n s + 39 n q - 156 n q s + 141 n q s 4 3 3 4 2 5 6 7 8 + 250 n q s - 655 n q s + 576 n q s - 229 n q s + 34 n s + 3 q 7 6 2 5 3 4 4 3 5 2 6 - 9 q s - 15 q s + 107 q s - 205 q s + 197 q s - 101 q s 7 8 7 6 6 5 2 5 + 25 q s - 2 s - 46 n - 208 n q + 94 n s - 360 n q + 192 n q s 5 2 4 3 4 2 4 2 4 3 3 4 + 90 n s - 274 n q - 156 n q s + 792 n q s - 378 n s - 46 n q 3 3 3 2 2 3 3 3 4 2 5 - 728 n q s + 1872 n q s - 1440 n q s + 342 n s + 60 n q 2 4 2 3 2 2 2 3 2 4 2 5 - 738 n q s + 1860 n q s - 1848 n q s + 768 n q s - 102 n s 6 5 4 2 3 3 2 4 + 36 n q - 312 n q s + 822 n q s - 952 n q s + 504 n q s 5 6 7 6 5 2 4 3 3 4 - 96 n q s - 2 n s + 6 q - 48 q s + 132 q s - 166 q s + 94 q s 2 5 6 7 6 5 5 4 2 - 12 q s - 8 q s + 2 s - 108 n - 483 n q + 318 n s - 867 n q 4 4 2 3 3 3 2 3 2 + 1053 n q s - 258 n s - 789 n q + 1266 n q s - 426 n q s 3 3 2 4 2 3 2 3 2 4 5 - 60 n s - 375 n q + 633 n q s - 426 n q s + 168 n s - 84 n q 4 3 2 2 3 4 5 6 + 90 n q s + 273 n q s - 546 n q s + 333 n q s - 66 n s - 6 q 5 4 2 3 3 2 4 5 6 5 - 12 q s + 105 q s - 189 q s + 147 q s - 51 q s + 6 s - 74 n 4 4 3 2 3 3 2 2 3 - 310 n q + 250 n s - 506 n q + 784 n q s - 284 n s - 400 n q 2 2 2 2 2 3 4 3 + 882 n q s - 588 n q s + 108 n s - 152 n q + 416 n q s 2 2 3 4 5 4 3 2 2 3 - 366 n q s + 96 n q s + 6 n s - 22 q + 68 q s - 64 q s + 6 q s 4 5 4 3 3 2 2 2 + 18 q s - 6 s + 4 n - 17 n q + 50 n s - 57 n q + 177 n q s 2 2 3 2 2 3 4 - 102 n s - 49 n q + 180 n q s - 183 n q s + 54 n s - 13 q 3 2 2 3 4 3 2 2 + 55 q s - 75 q s + 39 q s - 6 s + 30 n + 60 n q - 30 n s 2 2 3 2 3 2 + 36 n q - 24 n q s - 6 n s + 6 q - 12 q s + 6 s + 14 n + 22 n q / 2 \ 2 2 |d | / - 16 n s + 8 q - 10 q s + 2 s + 2 n + 2 q - 2 s) |--- A[n](r, s)| / ( | 2 | / \ds / 2 6 5 2 5 4 3 (n + n q - n s + q - s - 1) (2 n q + 8 n q - 12 n q s + 10 n q 4 2 4 2 3 3 3 2 2 3 3 - 40 n q s + 30 n q s - 40 n q s + 80 n q s - 40 n q s 2 5 2 3 2 2 2 3 2 4 6 5 - 10 n q + 60 n q s - 80 n q s + 30 n q s - 8 n q + 20 n q s 3 3 2 4 5 7 6 5 2 - 40 n q s + 40 n q s - 12 n q s - 2 q + 8 q s - 10 q s 3 4 2 5 6 6 5 5 4 2 4 + 10 q s - 8 q s + 2 q s - n - 4 n q + 6 n s + n q + 20 n q s 4 2 3 3 3 2 3 2 3 3 2 4 - 15 n s + 24 n q - 4 n q s - 40 n q s + 20 n s + 41 n q 2 3 2 2 2 2 3 2 4 5 4 - 72 n q s + 6 n q s + 40 n q s - 15 n s + 28 n q - 82 n q s 3 2 2 3 4 5 6 5 4 2 + 72 n q s - 4 n q s - 20 n q s + 6 n s + 7 q - 28 q s + 41 q s 3 3 2 4 5 6 5 4 4 3 2 - 24 q s + q s + 4 q s - s - n - 13 n q + 5 n s - 36 n q 3 3 2 2 3 2 2 2 2 2 3 + 52 n q s - 10 n s - 40 n q + 108 n q s - 78 n q s + 10 n s 4 3 2 2 3 4 5 4 - 19 n q + 80 n q s - 108 n q s + 52 n q s - 5 n s - 3 q + 19 q s 3 2 2 3 4 5 4 3 3 2 2 - 40 q s + 36 q s - 13 q s + s + 4 n + 4 n q - 16 n s - 13 n q 2 2 2 3 2 2 3 4 - 12 n q s + 24 n s - 22 n q + 26 n q s + 12 n q s - 16 n s - 9 q 3 2 2 3 4 3 2 2 2 + 22 q s - 13 q s - 4 q s + 4 s + 6 n + 20 n q - 18 n s + 17 n q 2 3 2 2 3 2 - 40 n q s + 18 n s + 3 q - 17 q s + 20 q s - 6 s - n + 8 n q 2 2 9 8 8 + 2 n s + 7 q - 8 q s - s - 5 n - q + 5 s - 2)) - (n + 6 n q - 3 n s 7 2 7 6 3 6 2 6 2 6 3 + 15 n q - 12 n q s + 20 n q - 15 n q s - 12 n q s + 8 n s 5 4 5 2 2 5 3 5 4 4 5 4 4 + 15 n q - 45 n q s + 36 n q s - 6 n s + 6 n q + 15 n q s 4 3 2 4 2 3 4 5 3 6 3 5 3 4 2 - 60 n q s + 45 n q s - 6 n s + n q + 12 n q s - 30 n q s 3 2 4 3 5 3 6 2 6 2 4 3 + 45 n q s - 36 n q s + 8 n s + 3 n q s - 30 n q s 2 3 4 2 2 5 2 6 6 2 5 3 + 60 n q s - 45 n q s + 12 n q s + 3 n q s - 12 n q s 4 4 2 6 7 8 6 3 5 4 + 15 n q s - 15 n q s + 12 n q s - 3 n s + q s - 6 q s 4 5 3 6 2 7 8 9 8 7 7 + 15 q s - 20 q s + 15 q s - 6 q s + s + 7 n + 38 n q - 20 n s 6 2 6 6 2 5 3 5 2 5 2 + 85 n q - 74 n q s + 4 n s + 100 n q - 90 n q s - 42 n q s 5 3 4 4 4 3 4 2 2 4 3 4 4 + 36 n s + 65 n q - 20 n q s - 165 n q s + 150 n q s - 30 n s 3 5 3 4 3 3 2 3 2 3 3 4 + 22 n q + 40 n q s - 200 n q s + 180 n q s - 30 n q s 3 5 2 6 2 5 2 4 2 2 3 3 - 12 n s + 3 n q + 30 n q s - 90 n q s + 40 n q s 2 2 4 2 5 2 6 6 5 2 4 3 + 75 n q s - 78 n q s + 20 n s + 6 n q s - 6 n q s - 40 n q s 3 4 2 5 6 7 6 2 5 3 + 100 n q s - 90 n q s + 34 n q s - 4 n s + 3 q s - 14 q s 4 4 3 5 2 6 7 8 7 6 6 + 25 q s - 20 q s + 5 q s + 2 q s - s + 21 n + 102 n q - 57 n s 5 2 5 5 2 4 3 4 2 4 2 + 201 n q - 192 n q s + 21 n s + 204 n q - 219 n q s - 42 n q s 4 3 3 4 3 3 3 2 2 3 3 + 63 n s + 111 n q - 72 n q s - 222 n q s + 240 n q s 3 4 2 5 2 4 2 3 2 2 2 3 - 57 n s + 30 n q + 33 n q s - 240 n q s + 258 n q s 2 4 2 5 6 5 4 2 3 3 - 78 n q s - 3 n s + 3 n q + 24 n q s - 87 n q s + 72 n q s 2 4 5 6 6 5 2 4 3 3 4 + 21 n q s - 48 n q s + 15 n s + 3 q s - 6 q s - 9 q s + 36 q s 2 5 6 7 6 5 5 4 2 - 39 q s + 18 q s - 3 s + 35 n + 150 n q - 90 n s + 255 n q 4 4 2 3 3 3 2 3 2 3 3 - 270 n q s + 45 n s + 216 n q - 276 n q s + 12 n q s + 52 n s 2 4 2 3 2 2 2 2 3 2 4 5 + 93 n q - 96 n q s - 126 n q s + 180 n q s - 51 n s + 18 n q 4 3 2 2 3 4 5 6 5 + 6 n q s - 120 n q s + 156 n q s - 66 n q s + 6 n s + q + 6 q s 4 2 3 3 2 4 5 6 5 4 - 27 q s + 32 q s - 9 q s - 6 q s + 3 s + 35 n + 130 n q 4 3 2 3 3 2 2 3 2 2 - 85 n s + 185 n q - 220 n q s + 50 n s + 124 n q - 189 n q s 2 2 2 3 4 3 2 2 3 + 48 n q s + 18 n s + 38 n q - 56 n q s - 21 n q s + 60 n q s 4 5 4 3 2 2 3 4 5 4 - 21 n s + 4 q - 2 q s - 20 q s + 33 q s - 18 q s + 3 s + 21 n 3 3 2 2 2 2 2 3 + 66 n q - 48 n s + 75 n q - 102 n q s + 30 n s + 36 n q 2 2 4 3 2 2 3 4 3 - 66 n q s + 30 n q s + 6 q - 12 q s + 3 q s + 6 q s - 3 s + 7 n 2 2 2 2 3 2 2 + 18 n q - 15 n s + 15 n q - 24 n q s + 9 n s + 4 q - 9 q s + 6 q s / 3 \ 3 2 2 2 |d | / 3 - s + n + 2 n q - 2 n s + q - 2 q s + s ) |--- A[n](r, s)| / ((n | 3 | / \ds / 2 2 2 2 2 2 + 2 n q - 2 n s + n q - 2 n q s + n s - 2 n - n q + n s + q - 2 q s 2 3 2 2 2 + s - n - 3 q + 3 s + 2) %1) + (n + 3 n q - 3 n s + 3 n q - 6 n q s 2 3 2 2 3 2 + 3 n s + q - 3 q s + 3 q s - s ) q A[n + 1](r, s)/(%1) /d \ 7 6 6 5 2 5 + |-- A[n + 1](r, s)| + (n + n q - 7 n s - 9 n q - 6 n q s \ds / 5 2 4 3 4 2 4 2 4 3 3 4 + 21 n s - 25 n q + 45 n q s + 15 n q s - 35 n s - 25 n q 3 3 3 2 2 3 3 3 4 2 5 2 4 + 100 n q s - 90 n q s - 20 n q s + 35 n s - 9 n q + 75 n q s 2 3 2 2 2 3 2 4 2 5 6 5 - 150 n q s + 90 n q s + 15 n q s - 21 n s + n q + 18 n q s 4 2 3 3 2 4 5 6 7 6 - 75 n q s + 100 n q s - 45 n q s - 6 n q s + 7 n s + q - q s 5 2 4 3 3 4 2 5 6 7 6 5 - 9 q s + 25 q s - 25 q s + 9 q s + q s - s + 7 n + 24 n q 5 4 2 4 4 2 3 3 3 2 - 42 n s + 21 n q - 120 n q s + 105 n s - 16 n q - 84 n q s 3 2 3 3 2 4 2 3 2 2 2 + 240 n q s - 140 n s - 39 n q + 48 n q s + 126 n q s 2 3 2 4 5 4 3 2 2 3 - 240 n q s + 105 n s - 24 n q + 78 n q s - 48 n q s - 84 n q s 4 5 6 5 4 2 3 3 2 4 + 120 n q s - 42 n s - 5 q + 24 q s - 39 q s + 16 q s + 21 q s 5 6 5 4 4 3 2 3 - 24 q s + 7 s + 15 n + 61 n q - 75 n s + 91 n q - 244 n q s 3 2 2 3 2 2 2 2 2 3 4 + 150 n s + 57 n q - 273 n q s + 366 n q s - 150 n s + 10 n q 3 2 2 3 4 5 4 - 114 n q s + 273 n q s - 244 n q s + 75 n s - 2 q - 10 q s 3 2 2 3 4 5 4 3 3 + 57 q s - 91 q s + 61 q s - 15 s + 11 n + 48 n q - 44 n s 2 2 2 2 2 3 2 2 + 75 n q - 144 n q s + 66 n s + 50 n q - 150 n q s + 144 n q s 3 4 3 2 2 3 4 3 2 - 44 n s + 12 q - 50 q s + 75 q s - 48 q s + 11 s - n + 3 n q 2 2 2 3 2 2 3 + 3 n s + 10 n q - 6 n q s - 3 n s + 6 q - 10 q s + 3 q s + s 2 2 2 - 3 n - 8 n q + 6 n s - 4 q + 8 q s - 3 s + n - q - s + 1) / 2 \ |d | / 5 4 2 4 |--- A[n + 1](r, s)| / ((-s - 1 + n + q) (2 n q + 6 n q - 10 n q s | 2 | / \ds / 3 3 3 2 3 2 2 4 2 3 2 2 2 + 4 n q - 24 n q s + 20 n q s - 4 n q - 12 n q s + 36 n q s 2 3 5 4 3 2 2 3 4 - 20 n q s - 6 n q + 8 n q s + 12 n q s - 24 n q s + 10 n q s 6 5 4 2 3 3 2 4 5 5 4 - 2 q + 6 q s - 4 q s - 4 q s + 6 q s - 2 q s - n + n q 4 3 2 3 3 2 2 3 2 2 + 5 n s + 12 n q - 4 n q s - 10 n s + 20 n q - 36 n q s 2 2 2 3 4 3 2 2 3 + 6 n q s + 10 n s + 13 n q - 40 n q s + 36 n q s - 4 n q s 4 5 4 3 2 2 3 4 5 4 - 5 n s + 3 q - 13 q s + 20 q s - 12 q s + q s + s - 3 n 3 3 2 2 2 2 2 3 2 - 8 n q + 12 n s - 4 n q + 24 n q s - 18 n s + 4 n q + 8 n q s 2 3 4 3 2 2 3 4 3 - 24 n q s + 12 n s + 3 q - 4 q s - 4 q s + 8 q s - 3 s - 2 n 2 2 2 2 3 2 - 10 n q + 6 n s - 11 n q + 20 n q s - 6 n s - 3 q + 11 q s 2 3 2 2 2 - 10 q s + 2 s + 2 n - 2 n q - 4 n s - 3 q + 2 q s + 2 s + 3 n + q 8 7 7 6 2 6 6 2 - 3 s + 1)) - 2 (n + 5 n q - 8 n s + 9 n q - 35 n q s + 28 n s 5 3 5 2 5 2 5 3 4 4 4 3 + 5 n q - 54 n q s + 105 n q s - 56 n s - 5 n q - 25 n q s 4 2 2 4 3 4 4 3 5 3 4 + 135 n q s - 175 n q s + 70 n s - 9 n q + 20 n q s 3 3 2 3 2 3 3 4 3 5 2 6 + 50 n q s - 180 n q s + 175 n q s - 56 n s - 5 n q 2 5 2 4 2 2 3 3 2 2 4 2 5 + 27 n q s - 30 n q s - 50 n q s + 135 n q s - 105 n q s 2 6 7 6 5 2 4 3 3 4 + 28 n s - n q + 10 n q s - 27 n q s + 20 n q s + 25 n q s 2 5 6 7 7 6 2 5 3 4 4 - 54 n q s + 35 n q s - 8 n s + q s - 5 q s + 9 q s - 5 q s 3 5 2 6 7 8 7 6 6 5 2 - 5 q s + 9 q s - 5 q s + s + 4 n + 22 n q - 28 n s + 49 n q 5 5 2 4 3 4 2 4 2 4 3 - 132 n q s + 84 n s + 55 n q - 245 n q s + 330 n q s - 140 n s 3 4 3 3 3 2 2 3 3 3 4 + 30 n q - 220 n q s + 490 n q s - 440 n q s + 140 n s 2 5 2 4 2 3 2 2 2 3 2 4 + 4 n q - 90 n q s + 330 n q s - 490 n q s + 330 n q s 2 5 6 5 4 2 3 3 2 4 - 84 n s - 3 n q - 8 n q s + 90 n q s - 220 n q s + 245 n q s 5 6 7 6 5 2 4 3 3 4 - 132 n q s + 28 n s - q + 3 q s + 4 q s - 30 q s + 55 q s 2 5 6 7 6 5 5 4 2 - 49 q s + 22 q s - 4 s + 3 n + 22 n q - 18 n s + 60 n q 4 4 2 3 3 3 2 3 2 3 3 - 110 n q s + 45 n s + 80 n q - 240 n q s + 220 n q s - 60 n s 2 4 2 3 2 2 2 2 3 2 4 5 + 55 n q - 240 n q s + 360 n q s - 220 n q s + 45 n s + 18 n q 4 3 2 2 3 4 5 6 - 110 n q s + 240 n q s - 240 n q s + 110 n q s - 18 n s + 2 q 5 4 2 3 3 2 4 5 6 5 - 18 q s + 55 q s - 80 q s + 60 q s - 22 q s + 3 s - 7 n 4 4 3 2 3 3 2 2 3 - 19 n q + 35 n s - 10 n q + 76 n q s - 70 n s + 14 n q 2 2 2 2 2 3 4 3 2 2 + 30 n q s - 114 n q s + 70 n s + 17 n q - 28 n q s - 30 n q s 3 4 5 4 3 2 2 3 4 + 76 n q s - 35 n s + 5 q - 17 q s + 14 q s + 10 q s - 19 q s 5 4 3 3 2 2 2 2 2 + 7 s - 13 n - 43 n q + 52 n s - 50 n q + 129 n q s - 78 n s 3 2 2 3 4 3 2 2 - 23 n q + 100 n q s - 129 n q s + 52 n s - 3 q + 23 q s - 50 q s 3 4 3 2 2 2 + 43 q s - 13 s - 6 n - 20 n q + 18 n s - 21 n q + 40 n q s 2 3 2 2 3 2 2 2 - 18 n s - 7 q + 21 q s - 20 q s + 6 s + n - 2 n s - q + s + n + q / 3 \ |d | / 6 5 2 5 4 3 - s) |--- A[n + 1](r, s)| / ((2 n q + 8 n q - 12 n q s + 10 n q | 3 | / \ds / 4 2 4 2 3 3 3 2 2 3 3 - 40 n q s + 30 n q s - 40 n q s + 80 n q s - 40 n q s 2 5 2 3 2 2 2 3 2 4 6 5 - 10 n q + 60 n q s - 80 n q s + 30 n q s - 8 n q + 20 n q s 3 3 2 4 5 7 6 5 2 - 40 n q s + 40 n q s - 12 n q s - 2 q + 8 q s - 10 q s 3 4 2 5 6 6 5 5 4 2 4 + 10 q s - 8 q s + 2 q s - n - 4 n q + 6 n s + n q + 20 n q s 4 2 3 3 3 2 3 2 3 3 2 4 - 15 n s + 24 n q - 4 n q s - 40 n q s + 20 n s + 41 n q 2 3 2 2 2 2 3 2 4 5 4 - 72 n q s + 6 n q s + 40 n q s - 15 n s + 28 n q - 82 n q s 3 2 2 3 4 5 6 5 4 2 + 72 n q s - 4 n q s - 20 n q s + 6 n s + 7 q - 28 q s + 41 q s 3 3 2 4 5 6 5 4 4 3 2 - 24 q s + q s + 4 q s - s - n - 13 n q + 5 n s - 36 n q 3 3 2 2 3 2 2 2 2 2 3 + 52 n q s - 10 n s - 40 n q + 108 n q s - 78 n q s + 10 n s 4 3 2 2 3 4 5 4 - 19 n q + 80 n q s - 108 n q s + 52 n q s - 5 n s - 3 q + 19 q s 3 2 2 3 4 5 4 3 3 2 2 - 40 q s + 36 q s - 13 q s + s + 4 n + 4 n q - 16 n s - 13 n q 2 2 2 3 2 2 3 4 - 12 n q s + 24 n s - 22 n q + 26 n q s + 12 n q s - 16 n s - 9 q 3 2 2 3 4 3 2 2 2 + 22 q s - 13 q s - 4 q s + 4 s + 6 n + 20 n q - 18 n s + 17 n q 2 3 2 2 3 2 - 40 n q s + 18 n s + 3 q - 17 q s + 20 q s - 6 s - n + 8 n q 2 2 7 + 2 n s + 7 q - 8 q s - s - 5 n - q + 5 s - 2) (-s - 1 + n + q)) + (n 6 6 5 2 5 5 2 4 3 + 5 n q - 7 n s + 10 n q - 30 n q s + 21 n s + 10 n q 4 2 4 2 4 3 3 4 3 3 3 2 2 - 50 n q s + 75 n q s - 35 n s + 5 n q - 40 n q s + 100 n q s 3 3 3 4 2 5 2 4 2 3 2 2 2 3 - 100 n q s + 35 n s + n q - 15 n q s + 60 n q s - 100 n q s 2 4 2 5 5 4 2 3 3 2 4 + 75 n q s - 21 n s - 2 n q s + 15 n q s - 40 n q s + 50 n q s 5 6 5 2 4 3 3 4 2 5 6 7 - 30 n q s + 7 n s + q s - 5 q s + 10 q s - 10 q s + 5 q s - s 6 5 5 4 2 4 4 2 3 3 + 5 n + 22 n q - 30 n s + 38 n q - 110 n q s + 75 n s + 32 n q 3 2 3 2 3 3 2 4 2 3 - 152 n q s + 220 n q s - 100 n s + 13 n q - 96 n q s 2 2 2 2 3 2 4 5 4 3 2 + 228 n q s - 220 n q s + 75 n s + 2 n q - 26 n q s + 96 n q s 2 3 4 5 5 4 2 3 3 - 152 n q s + 110 n q s - 30 n s - 2 q s + 13 q s - 32 q s 2 4 5 6 5 4 4 3 2 + 38 q s - 22 q s + 5 s + 10 n + 38 n q - 50 n s + 55 n q 3 3 2 2 3 2 2 2 2 - 152 n q s + 100 n s + 37 n q - 165 n q s + 228 n q s 2 3 4 3 2 2 3 4 - 100 n s + 11 n q - 74 n q s + 165 n q s - 152 n q s + 50 n s 5 4 3 2 2 3 4 5 4 3 + q - 11 q s + 37 q s - 55 q s + 38 q s - 10 s + 10 n + 32 n q 3 2 2 2 2 2 3 2 - 40 n s + 37 n q - 96 n q s + 60 n s + 18 n q - 74 n q s 2 3 4 3 2 2 3 4 3 + 96 n q s - 40 n s + 3 q - 18 q s + 37 q s - 32 q s + 10 s + 5 n 2 2 2 2 3 2 + 13 n q - 15 n s + 11 n q - 26 n q s + 15 n s + 3 q - 11 q s 2 3 2 2 2 + 13 q s - 5 s + n + 2 n q - 2 n s + q - 2 q s + s ) / 4 \ |d | / 6 5 2 5 4 3 |--- A[n + 1](r, s)| / (2 n q + 8 n q - 12 n q s + 10 n q | 4 | / \ds / 4 2 4 2 3 3 3 2 2 3 3 - 40 n q s + 30 n q s - 40 n q s + 80 n q s - 40 n q s 2 5 2 3 2 2 2 3 2 4 6 5 - 10 n q + 60 n q s - 80 n q s + 30 n q s - 8 n q + 20 n q s 3 3 2 4 5 7 6 5 2 - 40 n q s + 40 n q s - 12 n q s - 2 q + 8 q s - 10 q s 3 4 2 5 6 6 5 5 4 2 4 + 10 q s - 8 q s + 2 q s - n - 4 n q + 6 n s + n q + 20 n q s 4 2 3 3 3 2 3 2 3 3 2 4 - 15 n s + 24 n q - 4 n q s - 40 n q s + 20 n s + 41 n q 2 3 2 2 2 2 3 2 4 5 4 - 72 n q s + 6 n q s + 40 n q s - 15 n s + 28 n q - 82 n q s 3 2 2 3 4 5 6 5 4 2 + 72 n q s - 4 n q s - 20 n q s + 6 n s + 7 q - 28 q s + 41 q s 3 3 2 4 5 6 5 4 4 3 2 - 24 q s + q s + 4 q s - s - n - 13 n q + 5 n s - 36 n q 3 3 2 2 3 2 2 2 2 2 3 + 52 n q s - 10 n s - 40 n q + 108 n q s - 78 n q s + 10 n s 4 3 2 2 3 4 5 4 - 19 n q + 80 n q s - 108 n q s + 52 n q s - 5 n s - 3 q + 19 q s 3 2 2 3 4 5 4 3 3 2 2 - 40 q s + 36 q s - 13 q s + s + 4 n + 4 n q - 16 n s - 13 n q 2 2 2 3 2 2 3 4 - 12 n q s + 24 n s - 22 n q + 26 n q s + 12 n q s - 16 n s - 9 q 3 2 2 3 4 3 2 2 2 + 22 q s - 13 q s - 4 q s + 4 s + 6 n + 20 n q - 18 n s + 17 n q 2 3 2 2 3 2 - 40 n q s + 18 n s + 3 q - 17 q s + 20 q s - 6 s - n + 8 n q 2 2 + 2 n s + 7 q - 8 q s - s - 5 n - q + 5 s - 2) = 0 4 3 2 3 2 2 2 2 4 %1 := 2 n q + 4 n q - 8 n q s - 12 n q s + 12 n q s - 4 n q 2 2 3 5 4 2 3 4 4 3 + 12 n q s - 8 n q s - 2 q + 4 q s - 4 q s + 2 q s - n + 4 n q 3 2 2 2 2 2 3 2 2 + 4 n s + 12 n q - 12 n q s - 6 n s + 8 n q - 24 n q s + 12 n q s 3 4 3 2 2 3 4 3 2 2 + 4 n s + q - 8 q s + 12 q s - 4 q s - s - 4 n + 12 n s + 8 n q 2 3 2 3 2 2 - 12 n s + 4 q - 8 q s + 4 s - 6 n - 4 n q + 12 n s + q + 4 q s 2 - 6 s - 4 n - 2 q + 4 s - 1 and in Maple notation -(n^3*p^4-3*n^3*p^3*r+n^3*p^3*s+3*n^3*p^2*r^2-3*n^3*p^2*r*s-n^3*p*r^3+3*n^3*p*r ^2*s-n^3*r^3*s-3*n^2*p^5+9*n^2*p^4*r-3*n^2*p^4*s-9*n^2*p^3*r^2+9*n^2*p^3*r*s+3* n^2*p^2*r^3-9*n^2*p^2*r^2*s+3*n^2*p*r^3*s+3*n*p^6-9*n*p^5*r+3*n*p^5*s+9*n*p^4*r ^2-9*n*p^4*r*s-3*n*p^3*r^3+9*n*p^3*r^2*s-3*n*p^2*r^3*s-p^7+3*p^6*r-p^6*s-3*p^5* r^2+3*p^5*r*s+p^4*r^3-3*p^4*r^2*s+p^3*r^3*s-10*n^3*p^3+21*n^3*p^2*r-9*n^3*p^2*s -12*n^3*p*r^2+18*n^3*p*r*s+n^3*r^3-9*n^3*r^2*s+36*n^2*p^4-81*n^2*p^3*r+33*n^2*p ^3*s+54*n^2*p^2*r^2-72*n^2*p^2*r*s-9*n^2*p*r^3+45*n^2*p*r^2*s-6*n^2*r^3*s-42*n* p^5+99*n*p^4*r-39*n*p^4*s-72*n*p^3*r^2+90*n*p^3*r*s+15*n*p^2*r^3-63*n*p^2*r^2*s +12*n*p*r^3*s+16*p^6-39*p^5*r+15*p^5*s+30*p^4*r^2-36*p^4*r*s-7*p^3*r^3+27*p^3*r ^2*s-6*p^2*r^3*s+35*n^3*p^2-44*n^3*p*r+26*n^3*p*s+9*n^3*r^2-26*n^3*r*s-165*n^2* p^3+258*n^2*p^2*r-132*n^2*p^2*s-99*n^2*p*r^2+186*n^2*p*r*s+6*n^2*r^3-54*n^2*r^2 *s+237*n*p^4-420*n*p^3*r+198*n*p^3*s+207*n*p^2*r^2-330*n*p^2*r*s-24*n*p*r^3+144 *n*p*r^2*s-12*n*r^3*s-107*p^5+206*p^4*r-92*p^4*s-117*p^3*r^2+170*p^3*r*s+18*p^2 *r^3-90*p^2*r^2*s+12*p*r^3*s-50*n^3*p+26*n^3*r-24*n^3*s+360*n^2*p^2-342*n^2*p*r +228*n^2*p*s+54*n^2*r^2-156*n^2*r*s-690*n*p^3+858*n*p^2*r-492*n*p^2*s-252*n*p*r ^2+528*n*p*r*s+12*n*r^3-108*n*r^2*s+388*p^4-566*p^3*r+296*p^3*s+222*p^2*r^2-396 *p^2*r*s-20*p*r^3+132*p*r^2*s-8*r^3*s+24*n^3-372*n^2*p+156*n^2*r-144*n^2*s+1092 *n*p^2-840*n*p*r+600*n*p*s+108*n*r^2-312*n*r*s-824*p^3+852*p^2*r-528*p^2*s-204* p*r^2+456*p*r*s+8*r^3-72*r^2*s+144*n^2-888*n*p+312*n*r-288*n*s+1024*p^2-664*p*r +496*p*s+72*r^2-208*r*s+288*n-688*p+208*r-192*s+192)/(n+1)/(n^2+2*n*r+r^2+2*n+2 *r+1)/(p-r-1)^3*A[n](r,s)+(3*n^3*p^6-14*n^3*p^5*r+4*n^3*p^5*s+25*n^3*p^4*r^2-20 *n^3*p^4*r*s-20*n^3*p^3*r^3+40*n^3*p^3*r^2*s+5*n^3*p^2*r^4-40*n^3*p^2*r^3*s+2*n ^3*p*r^5+20*n^3*p*r^4*s-n^3*r^6-4*n^3*r^5*s-6*n^2*p^7+24*n^2*p^6*r-9*n^2*p^6*s-\ 30*n^2*p^5*r^2+42*n^2*p^5*r*s-75*n^2*p^4*r^2*s+30*n^2*p^3*r^4+60*n^2*p^3*r^3*s-\ 24*n^2*p^2*r^5-15*n^2*p^2*r^4*s+6*n^2*p*r^6-6*n^2*p*r^5*s+3*n^2*r^6*s+3*n*p^8-6 *n*p^7*r+6*n*p^7*s-15*n*p^6*r^2-24*n*p^6*r*s+60*n*p^5*r^3+30*n*p^5*r^2*s-75*n*p ^4*r^4+42*n*p^3*r^5-30*n*p^3*r^4*s-9*n*p^2*r^6+24*n*p^2*r^5*s-6*n*p*r^6*s-4*p^8 *r-p^8*s+20*p^7*r^2+2*p^7*r*s-40*p^6*r^3+5*p^6*r^2*s+40*p^5*r^4-20*p^5*r^3*s-20 *p^4*r^5+25*p^4*r^4*s+4*p^3*r^6-14*p^3*r^5*s+3*p^2*r^6*s-39*n^3*p^5+149*n^3*p^4 *r-46*n^3*p^4*s-206*n^3*p^3*r^2+184*n^3*p^3*r*s+114*n^3*p^2*r^3-276*n^3*p^2*r^2 *s-11*n^3*p*r^4+184*n^3*p*r^3*s-7*n^3*r^5-46*n^3*r^4*s+96*n^2*p^6-330*n^2*p^5*r +129*n^2*p^5*s+348*n^2*p^4*r^2-507*n^2*p^4*r*s-12*n^2*p^3*r^3+738*n^2*p^3*r^2*s -192*n^2*p^2*r^4-462*n^2*p^2*r^3*s+102*n^2*p*r^5+93*n^2*p*r^4*s-12*n^2*r^6+9*n^ 2*r^5*s-60*n*p^7+123*n*p^6*r-105*n*p^6*s+147*n*p^5*r^2+372*n*p^5*r*s-618*n*p^4* r^3-423*n*p^4*r^2*s+642*n*p^3*r^4+72*n*p^3*r^3*s-273*n*p^2*r^5+177*n*p^2*r^4*s+ 39*n*p*r^6-108*n*p*r^5*s+15*n*r^6*s+3*p^8+58*p^7*r+22*p^7*s-289*p^6*r^2-49*p^6* r*s+516*p^5*r^3-39*p^5*r^2*s-439*p^4*r^4+206*p^4*r^3*s+178*p^3*r^5-224*p^3*r^4* s-27*p^2*r^6+99*p^2*r^5*s-15*p*r^6*s+202*n^3*p^4-606*n^3*p^3*r+202*n^3*p^3*s+ 606*n^3*p^2*r^2-606*n^3*p^2*r*s-202*n^3*p*r^3+606*n^3*p*r^2*s-202*n^3*r^3*s-636 *n^2*p^5+1830*n^2*p^4*r-744*n^2*p^4*s-1566*n^2*p^3*r^2+2370*n^2*p^3*r*s+78*n^2* p^2*r^3-2646*n^2*p^2*r^2*s+402*n^2*p*r^4+1158*n^2*p*r^3*s-108*n^2*r^5-138*n^2*r ^4*s+507*n*p^6-1005*n*p^5*r+765*n*p^5*s-486*n*p^4*r^2-2337*n*p^4*r*s+2460*n*p^3 *r^3+2304*n*p^3*r^2*s-2019*n*p^2*r^4-540*n*p^2*r^3*s+585*n*p*r^5-309*n*p*r^4*s-\ 42*n*r^6+117*n*r^5*s-54*p^7-333*p^6*r-204*p^6*s+1731*p^5*r^2+459*p^5*r*s-2716*p ^4*r^3+21*p^4*r^2*s+1902*p^3*r^4-796*p^3*r^3*s-591*p^2*r^5+732*p^2*r^4*s+61*p*r ^6-231*p*r^5*s+19*r^6*s-535*n^3*p^3+1178*n^3*p^2*r-427*n^3*p^2*s-751*n^3*p*r^2+ 854*n^3*p*r*s+108*n^3*r^3-427*n^3*r^2*s+2259*n^2*p^4-5220*n^2*p^3*r+2211*n^2*p^ 3*s+3387*n^2*p^2*r^2-5352*n^2*p^2*r*s-150*n^2*p*r^3+4071*n^2*p*r^2*s-276*n^2*r^ 4-930*n^2*r^3*s-2361*n*p^5+4284*n*p^4*r-3003*n*p^4*s+447*n*p^3*r^2+7590*n*p^3*r *s-4716*n*p^2*r^3-6033*n*p^2*r^2*s+2760*n*p*r^4+1308*n*p*r^3*s-414*n*r^5+138*n* r^4*s+414*p^6+919*p^5*r+1042*p^5*s-5543*p^4*r^2-2207*p^4*r*s+7448*p^3*r^3+619*p ^3*r^2*s-4059*p^2*r^4+1392*p^2*r^3*s+867*p*r^5-1023*p*r^4*s-46*r^6+177*r^5*s+ 767*n^3*p^2-1093*n^3*p*r+441*n^3*p*s+326*n^3*r^2-441*n^3*r*s-4644*n^2*p^3+8049* n^2*p^2*r-3582*n^2*p^2*s-3489*n^2*p*r^2+5841*n^2*p*r*s+84*n^2*r^3-2259*n^2*r^2* s+6621*n*p^4-10341*n*p^3*r+6855*n*p^3*s+762*n*p^2*r^2-13401*n*p^2*r*s+4338*n*p* r^3+7560*n*p*r^2*s-1380*n*r^4-1014*n*r^3*s-1764*p^5-1009*p^4*r-3208*p^4*s+10177 *p^3*r^2+5977*p^3*r*s-11186*p^2*r^3-2265*p^2*r^2*s+4256*p*r^4-1010*p*r^3*s-474* r^5+506*r^4*s-566*n^3*p+386*n^3*r-180*n^3*s+5529*n^2*p^2-6339*n^2*p*r+3021*n^2* p*s+1350*n^2*r^2-2481*n^2*r*s-11445*n*p^3+14166*n*p^2*r-9111*n*p^2*s-1737*n*p*r ^2+12180*n*p*r*s-1524*n*r^3-3609*n*r^2*s+4564*p^4-711*p^3*r+6100*p^3*s-10611*p^ 2*r^2-9189*p^2*r*s+8686*p*r^3+3099*p*r^2*s-1748*r^4+170*r^3*s+168*n^3-3534*n^2* p+1986*n^2*r-1044*n^2*s+11913*n*p^2-10203*n*p*r+6555*n*p*s+882*n*r^2-4467*n*r*s -7336*p^3+3087*p^2*r-7008*p^2*s+5745*p*r^2+7461*p*r*s-2708*r^3-1497*r^2*s+936*n ^2-6834*n*p+2982*n*r-1980*n*s+7151*p^2-3005*p*r+4463*p*s-1230*r^2-2483*r*s+1656 *n-3866*p+998*r-1212*s+888)/(n*p-n*r-n+p-r-1)/(p^2-2*p*r+r^2-3*p+3*r+2)^2/(n^2+ 2*n*r+r^2+2*n+2*r+1)*diff(A[n](r,s),r)-(3*n^3*p^6-12*n^3*p^5*r+6*n^3*p^5*s+15*n ^3*p^4*r^2-30*n^3*p^4*r*s+60*n^3*p^3*r^2*s-15*n^3*p^2*r^4-60*n^3*p^2*r^3*s+12*n ^3*p*r^5+30*n^3*p*r^4*s-3*n^3*r^6-6*n^3*r^5*s-3*n^2*p^7+3*n^2*p^6*r-9*n^2*p^6*s +27*n^2*p^5*r^2+36*n^2*p^5*r*s-75*n^2*p^4*r^3-45*n^2*p^4*r^2*s+75*n^2*p^3*r^4-\ 27*n^2*p^2*r^5+45*n^2*p^2*r^4*s-3*n^2*p*r^6-36*n^2*p*r^5*s+3*n^2*r^7+9*n^2*r^6* s+9*n*p^7*r+3*n*p^7*s-36*n*p^6*r^2-3*n*p^6*r*s+45*n*p^5*r^3-27*n*p^5*r^2*s+75*n *p^4*r^3*s-45*n*p^3*r^5-75*n*p^3*r^4*s+36*n*p^2*r^6+27*n*p^2*r^5*s-9*n*p*r^7+3* n*p*r^6*s-3*n*r^7*s-6*p^7*r^2-3*p^7*r*s+30*p^6*r^3+12*p^6*r^2*s-60*p^5*r^4-15*p ^5*r^3*s+60*p^4*r^5-30*p^3*r^6+15*p^3*r^5*s+6*p^2*r^7-12*p^2*r^6*s+3*p*r^7*s-42 *n^3*p^5+138*n^3*p^4*r-72*n^3*p^4*s-132*n^3*p^3*r^2+288*n^3*p^3*r*s-12*n^3*p^2* r^3-432*n^3*p^2*r^2*s+78*n^3*p*r^4+288*n^3*p*r^3*s-30*n^3*r^5-72*n^3*r^4*s+57*n ^2*p^6-72*n^2*p^5*r+144*n^2*p^5*s-279*n^2*p^4*r^2-504*n^2*p^4*r*s+696*n^2*p^3*r ^3+576*n^2*p^3*r^2*s-549*n^2*p^2*r^4-144*n^2*p^2*r^3*s+144*n^2*p*r^5-144*n^2*p* r^4*s+3*n^2*r^6+72*n^2*r^5*s-6*n*p^7-138*n*p^6*r-66*n*p^6*s+540*n*p^5*r^2+108*n *p^5*r*s-636*n*p^4*r^3+234*n*p^4*r^2*s+114*n*p^3*r^4-696*n*p^3*r^3*s+270*n*p^2* r^5+594*n*p^2*r^4*s-168*n*p*r^6-180*n*p*r^5*s+24*n*r^7+6*n*r^6*s+9*p^7*r+3*p^7* s+60*p^6*r^2+45*p^6*r*s-363*p^5*r^3-189*p^5*r^2*s+672*p^4*r^4+237*p^4*r^3*s-573 *p^3*r^5-63*p^3*r^4*s+228*p^2*r^6-81*p^2*r^5*s-33*p*r^7+57*p*r^6*s-9*r^7*s+236* n^3*p^4-611*n^3*p^3*r+333*n^3*p^3*s+417*n^3*p^2*r^2-999*n^3*p^2*r*s+55*n^3*p*r^ 3+999*n^3*p*r^2*s-97*n^3*r^4-333*n^3*r^3*s-444*n^2*p^5+588*n^2*p^4*r-924*n^2*p^ 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3036*n*s+3769*p^2+3371*p*r+5011*p*s-2880*r^2-1975*r*s+1656*n-2530*p-690*r-1564* s+696)/(n*p^2-2*n*p*r+n*r^2-3*n*p+3*n*r+p^2-2*p*r+r^2+2*n-3*p+3*r+2)/(p^3-3*p^2 *r+3*p*r^2-r^3-6*p^2+12*p*r-6*r^2+11*p-11*r-6)/(n^2+2*n*r+r^2+2*n+2*r+1)*diff( diff(A[n](r,s),r),r)+(n^3*p^4+4*n^3*p^3*s-6*n^3*p^2*r^2-12*n^3*p^2*r*s+8*n^3*p* r^3+12*n^3*p*r^2*s-3*n^3*r^4-4*n^3*r^3*s-6*n^2*p^4*r-3*n^2*p^4*s+12*n^2*p^3*r^2 +18*n^2*p^2*r^2*s-12*n^2*p*r^4-24*n^2*p*r^3*s+6*n^2*r^5+9*n^2*r^4*s+9*n*p^4*r^2 +6*n*p^4*r*s-24*n*p^3*r^3-12*n*p^3*r^2*s+18*n*p^2*r^4+12*n*p*r^4*s-3*n*r^6-6*n* r^5*s-4*p^4*r^3-3*p^4*r^2*s+12*p^3*r^4+8*p^3*r^3*s-12*p^2*r^5-6*p^2*r^4*s+4*p*r ^6+r^6*s-10*n^3*p^3-30*n^3*p^2*s+30*n^3*p*r^2+60*n^3*p*r*s-20*n^3*r^3-30*n^3*r^ 2*s+3*n^2*p^4+60*n^2*p^3*r+42*n^2*p^3*s-108*n^2*p^2*r^2-36*n^2*p^2*r*s+24*n^2*p *r^3-54*n^2*p*r^2*s+21*n^2*r^4+48*n^2*r^3*s-12*n*p^4*r-6*n*p^4*s-66*n*p^3*r^2-\ 60*n*p^3*r*s+180*n*p^2*r^3+126*n*p^2*r^2*s-114*n*p*r^4-48*n*p*r^3*s+12*n*r^5-12 *n*r^4*s+9*p^4*r^2+6*p^4*r*s+16*p^3*r^3+18*p^3*r^2*s-72*p^2*r^4-60*p^2*r^3*s+60 *p*r^5+42*p*r^4*s-13*r^6-6*r^5*s+35*n^3*p^2+70*n^3*p*s-35*n^3*r^2-70*n^3*r*s-30 *n^2*p^3-210*n^2*p^2*r-195*n^2*p^2*s+300*n^2*p*r^2+180*n^2*p*r*s-60*n^2*r^3+15* n^2*r^2*s+3*n*p^4+120*n*p^3*r+72*n*p^3*s+117*n*p^2*r^2+174*n*p^2*r*s-396*n*p*r^ 3-354*n*p*r^2*s+156*n*r^4+108*n*r^3*s-6*p^4*r-3*p^4*s-78*p^3*r^2-60*p^3*r*s+40* p^2*r^3+3*p^2*r^2*s+108*p*r^4+116*p*r^3*s-64*r^5-56*r^4*s-50*n^3*p-50*n^3*s+105 *n^2*p^2+300*n^2*p*r+360*n^2*p*s-255*n^2*r^2-210*n^2*r*s-30*n*p^3-420*n*p^2*r-\ 300*n*p^2*s+60*n*p*r^2-120*n*p*r*s+240*n*r^3+270*n*r^2*s+p^4+60*p^3*r+34*p^3*s+ 219*p^2*r^2+198*p^2*r*s-212*p*r^3-138*p*r^2*s-18*r^4-44*r^3*s+24*n^3-150*n^2*p-\ 144*n^2*r-222*n^2*s+105*n*p^2+600*n*p*r+510*n*p*s-189*n*r^2-66*n*r*s-10*p^3-210 *p^2*r-135*p^2*s-210*p*r^2-240*p*r*s+184*r^3+153*r^2*s+72*n^2-150*n*p-288*n*r-\ 294*n*s+35*p^2+300*p*r+220*p*s+31*r^2+74*r*s+72*n-50*p-144*r-122*s+24)/(n*p^3-3 *n*p^2*r+3*n*p*r^2-n*r^3-6*n*p^2+12*n*p*r-6*n*r^2+p^3-3*p^2*r+3*p*r^2-r^3+11*n* p-11*n*r-6*p^2+12*p*r-6*r^2-6*n+11*p-11*r-6)/(n+r+1)^2*diff(diff(diff(A[n](r,s) ,r),r),r)-(n^3*r+n^3*s-3*n^2*r^2-3*n^2*r*s+3*n*r^3+3*n*r^2*s-r^4-r^3*s+3*n^2*r+ 3*n^2*s-6*n*r^2-6*n*r*s+3*r^3+3*r^2*s+3*n*r+3*n*s-3*r^2-3*r*s+r+s)/(n^3+2*n^2*r +n*r^2+3*n^2+4*n*r+r^2+3*n+2*r+1)*diff(diff(diff(diff(A[n](r,s),r),r),r),r)+(n^ 2*p^3-3*n^2*p^2*r+3*n^2*p*r^2-n^2*r^3+2*n*p^4-6*n*p^3*r+6*n*p^2*r^2-2*n*p*r^3+p ^5-3*p^4*r+3*p^3*r^2-p^2*r^3-9*n^2*p^2+18*n^2*p*r-9*n^2*r^2-18*n*p^3+36*n*p^2*r -18*n*p*r^2-9*p^4+18*p^3*r-9*p^2*r^2+26*n^2*p-26*n^2*r+52*n*p^2-52*n*p*r+26*p^3 -26*p^2*r-24*n^2-48*n*p-24*p^2)/(n^2+2*n*r+r^2+2*n+2*r+1)/(p-r-1)^3*A[n+1](r,s) -(4*n^2*p^5-20*n^2*p^4*r+40*n^2*p^3*r^2-40*n^2*p^2*r^3+20*n^2*p*r^4-4*n^2*r^5+6 *n*p^6-28*n*p^5*r+50*n*p^4*r^2-40*n*p^3*r^3+10*n*p^2*r^4+4*n*p*r^5-2*n*r^6+2*p^ 7-8*p^6*r+10*p^5*r^2-10*p^3*r^4+8*p^2*r^5-2*p*r^6-46*n^2*p^4+184*n^2*p^3*r-276* n^2*p^2*r^2+184*n^2*p*r^3-46*n^2*r^4-70*n*p^5+258*n*p^4*r-332*n*p^3*r^2+148*n*p ^2*r^3+18*n*p*r^4-22*n*r^5-23*p^6+68*p^5*r-41*p^4*r^2-56*p^3*r^3+79*p^2*r^4-28* p*r^5+r^6+202*n^2*p^3-606*n^2*p^2*r+606*n^2*p*r^2-202*n^2*r^3+312*n*p^4-844*n*p ^3*r+660*n*p^2*r^2-36*n*p*r^3-92*n*r^4+99*p^5-183*p^4*r-56*p^3*r^2+276*p^2*r^3-\ 147*p*r^4+11*r^5-427*n^2*p^2+854*n^2*p*r-427*n^2*r^2-666*n*p^3+1144*n*p^2*r-290 *n*p*r^2-188*n*r^3-193*p^4+106*p^3*r+413*p^2*r^2-372*p*r^3+46*r^4+441*n^2*p-441 *n^2*r+680*n*p^2-478*n*p*r-202*n*r^2+145*p^3+245*p^2*r-484*p*r^2+94*r^3-180*n^2 -250*n*p-110*n*r+31*p^2-312*p*r+101*r^2-24*n-79*p+55*r+12)/(p-r-1)/(n+r+1)^2/(p ^2-2*p*r+r^2-3*p+3*r+2)^2*diff(A[n+1](r,s),r)+(6*n^2*p^5-30*n^2*p^4*r+60*n^2*p^ 3*r^2-60*n^2*p^2*r^3+30*n^2*p*r^4-6*n^2*r^5+6*n*p^6-24*n*p^5*r+30*n*p^4*r^2-30* n*p^2*r^4+24*n*p*r^5-6*n*r^6+p^7-p^6*r-9*p^5*r^2+25*p^4*r^3-25*p^3*r^4+9*p^2*r^ 5+p*r^6-r^7-72*n^2*p^4+288*n^2*p^3*r-432*n^2*p^2*r^2+288*n^2*p*r^3-72*n^2*r^4-\ 72*n*p^5+216*n*p^4*r-144*n*p^3*r^2-144*n*p^2*r^3+216*n*p*r^4-72*n*r^5-10*p^6-12 *p^5*r+138*p^4*r^2-232*p^3*r^3+138*p^2*r^4-12*p*r^5-10*r^6+333*n^2*p^3-999*n^2* p^2*r+999*n^2*p*r^2-333*n^2*r^3+328*n*p^4-646*n*p^3*r-30*n*p^2*r^2+686*n*p*r^3-\ 338*n*r^4+28*p^5+188*p^4*r-699*p^3*r^2+689*p^2*r^3-173*p*r^4-33*r^5-744*n^2*p^2 +1488*n^2*p*r-744*n^2*r^2-694*n*p^3+594*n*p^2*r+894*n*p*r^2-794*n*r^3+24*p^4-\ 790*p^3*r+1482*p^2*r^2-690*p*r^3-26*r^4+805*n^2*p-805*n^2*r+626*n*p^2+358*n*p*r -984*n*r^2-258*p^3+1400*p^2*r-1221*p*r^2+79*r^3-340*n^2-66*n*p-614*n*r+466*p^2-\ 998*p*r+192*r^2-152*n-307*p+155*r+44)/(p^2-2*p*r+r^2-3*p+3*r+2)/(p^3-3*p^2*r+3* p*r^2-r^3-6*p^2+12*p*r-6*r^2+11*p-11*r-6)/(n^2+2*n*r+r^2+2*n+2*r+1)*diff(diff(A [n+1](r,s),r),r)-2*(2*n*p^3-6*n*p^2*r+6*n*p*r^2-2*n*r^3+p^4-2*p^3*r+2*p*r^3-r^4 -15*n*p^2+30*n*p*r-15*n*r^2-8*p^3+9*p^2*r+6*p*r^2-7*r^3+35*n*p-35*n*r+20*p^2-5* p*r-15*r^2-25*n-15*p-10*r-1)/(p^3-3*p^2*r+3*p*r^2-r^3-6*p^2+12*p*r-6*r^2+11*p-\ 11*r-6)/(n+r+1)*diff(diff(diff(A[n+1](r,s),r),r),r)+diff(diff(diff(diff(A[n+1]( r,s),r),r),r),r) = 0 (8*n^7+44*n^6*q-32*n^6*s+102*n^5*q^2-144*n^5*q*s+48*n^5*s^2+129*n^4*q^3-264*n^4 *q^2*s+168*n^4*q*s^2-32*n^4*s^3+96*n^3*q^4-252*n^3*q^3*s+228*n^3*q^2*s^2-80*n^3 *q*s^3+8*n^3*s^4+42*n^2*q^5-132*n^2*q^4*s+150*n^2*q^3*s^2-72*n^2*q^2*s^3+12*n^2 *q*s^4+10*n*q^6-36*n*q^5*s+48*n*q^4*s^2-28*n*q^3*s^3+6*n*q^2*s^4+q^7-4*q^6*s+6* q^5*s^2-4*q^4*s^3+q^3*s^4+44*n^6+204*n^5*q-144*n^5*s+387*n^4*q^2-528*n^4*q*s+ 168*n^4*s^2+384*n^3*q^3-756*n^3*q^2*s+456*n^3*q*s^2-80*n^3*s^3+210*n^2*q^4-528* n^2*q^3*s+450*n^2*q^2*s^2-144*n^2*q*s^3+12*n^2*s^4+60*n*q^5-180*n*q^4*s+192*n*q ^3*s^2-84*n*q^2*s^3+12*n*q*s^4+7*q^6-24*q^5*s+30*q^4*s^2-16*q^3*s^3+3*q^2*s^4+ 102*n^5+387*n^4*q-264*n^4*s+576*n^3*q^2-756*n^3*q*s+228*n^3*s^2+420*n^2*q^3-792 *n^2*q^2*s+450*n^2*q*s^2-72*n^2*s^3+150*n*q^4-360*n*q^3*s+288*n*q^2*s^2-84*n*q* s^3+6*n*s^4+21*q^5-60*q^4*s+60*q^3*s^2-24*q^2*s^3+3*q*s^4+129*n^4+384*n^3*q-252 *n^3*s+420*n^2*q^2-528*n^2*q*s+150*n^2*s^2+200*n*q^3-360*n*q^2*s+192*n*q*s^2-28 *n*s^3+35*q^4-80*q^3*s+60*q^2*s^2-16*q*s^3+s^4+96*n^3+210*n^2*q-132*n^2*s+150*n *q^2-180*n*q*s+48*n*s^2+35*q^3-60*q^2*s+30*q*s^2-4*s^3+42*n^2+60*n*q-36*n*s+21* q^2-24*q*s+6*s^2+10*n+7*q-4*s+1)/(n+1)/(2*n^4*q+4*n^3*q^2-8*n^3*q*s-12*n^2*q^2* s+12*n^2*q*s^2-4*n*q^4+12*n*q^2*s^2-8*n*q*s^3-2*q^5+4*q^4*s-4*q^2*s^3+2*q*s^4-n ^4+4*n^3*q+4*n^3*s+12*n^2*q^2-12*n^2*q*s-6*n^2*s^2+8*n*q^3-24*n*q^2*s+12*n*q*s^ 2+4*n*s^3+q^4-8*q^3*s+12*q^2*s^2-4*q*s^3-s^4-4*n^3+12*n^2*s+8*n*q^2-12*n*s^2+4* q^3-8*q^2*s+4*s^3-6*n^2-4*n*q+12*n*s+q^2+4*q*s-6*s^2-4*n-2*q+4*s-1)*A[n](r,s)-( 12*n^9+84*n^8*q-60*n^8*s+255*n^7*q^2-348*n^7*q*s+108*n^7*s^2+438*n^6*q^3-843*n^ 6*q^2*s+468*n^6*q*s^2-60*n^6*s^3+465*n^5*q^4-1092*n^5*q^3*s+747*n^5*q^2*s^2-60* n^5*q*s^3-60*n^5*s^4+312*n^4*q^5-795*n^4*q^4*s+450*n^4*q^3*s^2+345*n^4*q^2*s^3-\ 420*n^4*q*s^4+108*n^4*s^5+129*n^3*q^6-300*n^3*q^5*s-90*n^3*q^4*s^2+840*n^3*q^3* s^3-915*n^3*q^2*s^4+396*n^3*q*s^5-60*n^3*s^6+30*n^2*q^7-33*n^2*q^6*s-252*n^2*q^ 5*s^2+750*n^2*q^4*s^3-870*n^2*q^3*s^4+495*n^2*q^2*s^5-132*n^2*q*s^6+12*n^2*s^7+ 3*n*q^8+12*n*q^7*s-117*n*q^6*s^2+300*n*q^5*s^3-375*n*q^4*s^4+252*n*q^3*s^5-87*n *q^2*s^6+12*n*q*s^7+3*q^8*s-18*q^7*s^2+45*q^6*s^3-60*q^5*s^4+45*q^4*s^5-18*q^3* s^6+3*q^2*s^7+54*n^8+339*n^7*q-246*n^7*s+912*n^6*q^2-1275*n^6*q*s+414*n^6*s^2+ 1368*n^5*q^3-2736*n^5*q^2*s+1647*n^5*q*s^2-270*n^5*s^3+1245*n^4*q^4-3120*n^4*q^ 3*s+2520*n^4*q^2*s^2-615*n^4*q*s^3-30*n^4*s^4+699*n^3*q^5-2010*n^3*q^4*s+1800*n ^3*q^3*s^2-240*n^3*q^2*s^3-375*n^3*q*s^4+126*n^3*s^5+234*n^2*q^6-711*n^2*q^5*s+ 540*n^2*q^4*s^2+360*n^2*q^3*s^3-720*n^2*q^2*s^4+351*n^2*q*s^5-54*n^2*s^6+42*n*q ^7-120*n*q^6*s+9*n*q^5*s^2+330*n*q^4*s^3-480*n*q^3*s^4+288*n*q^2*s^5-75*n*q*s^6 +6*n*s^7+3*q^8-6*q^7*s-18*q^6*s^2+75*q^5*s^3-105*q^4*s^4+72*q^3*s^5-24*q^2*s^6+ 3*q*s^7+85*n^7+478*n^6*q-361*n^6*s+1134*n^5*q^2-1668*n^5*q*s+585*n^5*s^2+1468*n ^4*q^3-3138*n^4*q^2*s+2106*n^4*q*s^2-429*n^4*s^3+1117*n^3*q^4-3064*n^3*q^3*s+ 2916*n^3*q^2*s^2-1080*n^3*q*s^3+111*n^3*s^4+498*n^2*q^5-1629*n^2*q^4*s+1920*n^2 *q^3*s^2-924*n^2*q^2*s^3+114*n^2*q*s^4+21*n^2*s^5+120*n*q^6-444*n*q^5*s+591*n*q ^4*s^2-296*n*q^3*s^3-18*n*q^2*s^4+60*n*q*s^5-13*n*s^6+12*q^7-48*q^6*s+66*q^5*s^ 2-23*q^4*s^3-28*q^3*s^4+30*q^2*s^5-10*q*s^6+s^7+37*n^6+201*n^5*q-180*n^5*s+447* n^4*q^2-783*n^4*q*s+333*n^4*s^2+520*n^3*q^3-1332*n^3*q^2*s+1098*n^3*q*s^2-288*n ^3*s^3+333*n^2*q^4-1104*n^2*q^3*s+1314*n^2*q^2*s^2-654*n^2*q*s^3+111*n^2*s^4+ 111*n*q^5-444*n*q^4*s+672*n*q^3*s^2-468*n*q^2*s^3+141*n*q*s^4-12*n*s^5+15*q^6-\ 69*q^5*s+123*q^4*s^2-104*q^3*s^3+39*q^2*s^4-3*q*s^5-s^6-39*n^5-132*n^4*q+69*n^4 *s-165*n^3*q^2+132*n^3*q*s+6*n^3*s^2-90*n^2*q^3+45*n^2*q^2*s+108*n^2*q*s^2-66*n ^2*s^3-18*n*q^4-36*n*q^3*s+153*n*q^2*s^2-132*n*q*s^3+33*n*s^4-18*q^4*s+54*q^3*s ^2-57*q^2*s^3+24*q*s^4-3*s^5-43*n^4-139*n^3*q+106*n^3*s-162*n^2*q^2+231*n^2*q*s -72*n^2*s^2-80*n*q^3+156*n*q^2*s-81*n*q*s^2+6*n*s^3-14*q^4+32*q^3*s-18*q^2*s^2-\ 3*q*s^3+3*s^4-n^3-18*n^2*q+33*n^2*s-24*n*q^2+60*n*q*s-27*n*s^2-8*q^3+24*q^2*s-\ 18*q*s^2+3*s^3+15*n^2+15*n*q+3*q^2+3*q*s-3*s^2+7*n+4*q-s+1)/(n+1)/(-s-1+n+q)/(2 *n^5*q+6*n^4*q^2-10*n^4*q*s+4*n^3*q^3-24*n^3*q^2*s+20*n^3*q*s^2-4*n^2*q^4-12*n^ 2*q^3*s+36*n^2*q^2*s^2-20*n^2*q*s^3-6*n*q^5+8*n*q^4*s+12*n*q^3*s^2-24*n*q^2*s^3 +10*n*q*s^4-2*q^6+6*q^5*s-4*q^4*s^2-4*q^3*s^3+6*q^2*s^4-2*q*s^5-n^5+n^4*q+5*n^4 *s+12*n^3*q^2-4*n^3*q*s-10*n^3*s^2+20*n^2*q^3-36*n^2*q^2*s+6*n^2*q*s^2+10*n^2*s ^3+13*n*q^4-40*n*q^3*s+36*n*q^2*s^2-4*n*q*s^3-5*n*s^4+3*q^5-13*q^4*s+20*q^3*s^2 -12*q^2*s^3+q*s^4+s^5-3*n^4-8*n^3*q+12*n^3*s-4*n^2*q^2+24*n^2*q*s-18*n^2*s^2+4* n*q^3+8*n*q^2*s-24*n*q*s^2+12*n*s^3+3*q^4-4*q^3*s-4*q^2*s^2+8*q*s^3-3*s^4-2*n^3 -10*n^2*q+6*n^2*s-11*n*q^2+20*n*q*s-6*n*s^2-3*q^3+11*q^2*s-10*q*s^2+2*s^3+2*n^2 -2*n*q-4*n*s-3*q^2+2*q*s+2*s^2+3*n+q-3*s+1)*diff(A[n](r,s),s)+(6*n^10+45*n^9*q-\ 30*n^9*s+147*n^8*q^2-183*n^8*q*s+48*n^8*s^2+273*n^7*q^3-462*n^7*q^2*s+192*n^7*q *s^2+315*n^6*q^4-609*n^6*q^3*s+210*n^6*q^2*s^2+168*n^6*q*s^3-84*n^6*s^4+231*n^5 *q^5-420*n^5*q^4*s-147*n^5*q^3*s^2+714*n^5*q^2*s^3-462*n^5*q*s^4+84*n^5*s^5+105 *n^4*q^6-105*n^4*q^5*s-525*n^4*q^4*s^2+1155*n^4*q^3*s^3-840*n^4*q^2*s^4+210*n^4 *q*s^5+27*n^3*q^7+42*n^3*q^6*s-462*n^3*q^5*s^2+840*n^3*q^4*s^3-525*n^3*q^3*s^4-\ 42*n^3*q^2*s^5+168*n^3*q*s^6-48*n^3*s^7+3*n^2*q^8+33*n^2*q^7*s-168*n^2*q^6*s^2+ 210*n^2*q^5*s^3+105*n^2*q^4*s^4-483*n^2*q^3*s^5+462*n^2*q^2*s^6-192*n^2*q*s^7+ 30*n^2*s^8+6*n*q^8*s-15*n*q^7*s^2-42*n*q^6*s^3+231*n*q^5*s^4-420*n*q^4*s^5+399* n*q^3*s^6-210*n*q^2*s^7+57*n*q*s^8-6*n*s^9+3*q^8*s^2-21*q^7*s^3+63*q^6*s^4-105* q^5*s^5+105*q^4*s^6-63*q^3*s^7+21*q^2*s^8-3*q*s^9+24*n^9+168*n^8*q-120*n^8*s+ 510*n^7*q^2-696*n^7*q*s+216*n^7*s^2+876*n^6*q^3-1686*n^6*q^2*s+936*n^6*q*s^2-\ 120*n^6*s^3+930*n^5*q^4-2184*n^5*q^3*s+1494*n^5*q^2*s^2-120*n^5*q*s^3-120*n^5*s 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6*n^2*s^2+8*n*q^3-24*n*q^2*s+12*n*q*s^2+4*n*s^3+q^4-8*q^3*s+12*q^2*s^2-4*q*s^3- s^4-4*n^3+12*n^2*s+8*n*q^2-12*n*s^2+4*q^3-8*q^2*s+4*s^3-6*n^2-4*n*q+12*n*s+q^2+ 4*q*s-6*s^2-4*n-2*q+4*s-1)*diff(diff(diff(A[n](r,s),s),s),s)+(n^3+3*n^2*q-3*n^2 *s+3*n*q^2-6*n*q*s+3*n*s^2+q^3-3*q^2*s+3*q*s^2-s^3)*q^2/(2*n^4*q+4*n^3*q^2-8*n^ 3*q*s-12*n^2*q^2*s+12*n^2*q*s^2-4*n*q^4+12*n*q^2*s^2-8*n*q*s^3-2*q^5+4*q^4*s-4* q^2*s^3+2*q*s^4-n^4+4*n^3*q+4*n^3*s+12*n^2*q^2-12*n^2*q*s-6*n^2*s^2+8*n*q^3-24* n*q^2*s+12*n*q*s^2+4*n*s^3+q^4-8*q^3*s+12*q^2*s^2-4*q*s^3-s^4-4*n^3+12*n^2*s+8* n*q^2-12*n*s^2+4*q^3-8*q^2*s+4*s^3-6*n^2-4*n*q+12*n*s+q^2+4*q*s-6*s^2-4*n-2*q+4 *s-1)*A[n+1](r,s)+diff(A[n+1](r,s),s)+(n^7+n^6*q-7*n^6*s-9*n^5*q^2-6*n^5*q*s+21 *n^5*s^2-25*n^4*q^3+45*n^4*q^2*s+15*n^4*q*s^2-35*n^4*s^3-25*n^3*q^4+100*n^3*q^3 *s-90*n^3*q^2*s^2-20*n^3*q*s^3+35*n^3*s^4-9*n^2*q^5+75*n^2*q^4*s-150*n^2*q^3*s^ 2+90*n^2*q^2*s^3+15*n^2*q*s^4-21*n^2*s^5+n*q^6+18*n*q^5*s-75*n*q^4*s^2+100*n*q^ 3*s^3-45*n*q^2*s^4-6*n*q*s^5+7*n*s^6+q^7-q^6*s-9*q^5*s^2+25*q^4*s^3-25*q^3*s^4+ 9*q^2*s^5+q*s^6-s^7+7*n^6+24*n^5*q-42*n^5*s+21*n^4*q^2-120*n^4*q*s+105*n^4*s^2-\ 16*n^3*q^3-84*n^3*q^2*s+240*n^3*q*s^2-140*n^3*s^3-39*n^2*q^4+48*n^2*q^3*s+126*n ^2*q^2*s^2-240*n^2*q*s^3+105*n^2*s^4-24*n*q^5+78*n*q^4*s-48*n*q^3*s^2-84*n*q^2* s^3+120*n*q*s^4-42*n*s^5-5*q^6+24*q^5*s-39*q^4*s^2+16*q^3*s^3+21*q^2*s^4-24*q*s ^5+7*s^6+15*n^5+61*n^4*q-75*n^4*s+91*n^3*q^2-244*n^3*q*s+150*n^3*s^2+57*n^2*q^3 -273*n^2*q^2*s+366*n^2*q*s^2-150*n^2*s^3+10*n*q^4-114*n*q^3*s+273*n*q^2*s^2-244 *n*q*s^3+75*n*s^4-2*q^5-10*q^4*s+57*q^3*s^2-91*q^2*s^3+61*q*s^4-15*s^5+11*n^4+ 48*n^3*q-44*n^3*s+75*n^2*q^2-144*n^2*q*s+66*n^2*s^2+50*n*q^3-150*n*q^2*s+144*n* q*s^2-44*n*s^3+12*q^4-50*q^3*s+75*q^2*s^2-48*q*s^3+11*s^4-n^3+3*n^2*q+3*n^2*s+ 10*n*q^2-6*n*q*s-3*n*s^2+6*q^3-10*q^2*s+3*q*s^2+s^3-3*n^2-8*n*q+6*n*s-4*q^2+8*q *s-3*s^2+n-q-s+1)/(-s-1+n+q)/(2*n^5*q+6*n^4*q^2-10*n^4*q*s+4*n^3*q^3-24*n^3*q^2 *s+20*n^3*q*s^2-4*n^2*q^4-12*n^2*q^3*s+36*n^2*q^2*s^2-20*n^2*q*s^3-6*n*q^5+8*n* q^4*s+12*n*q^3*s^2-24*n*q^2*s^3+10*n*q*s^4-2*q^6+6*q^5*s-4*q^4*s^2-4*q^3*s^3+6* q^2*s^4-2*q*s^5-n^5+n^4*q+5*n^4*s+12*n^3*q^2-4*n^3*q*s-10*n^3*s^2+20*n^2*q^3-36 *n^2*q^2*s+6*n^2*q*s^2+10*n^2*s^3+13*n*q^4-40*n*q^3*s+36*n*q^2*s^2-4*n*q*s^3-5* n*s^4+3*q^5-13*q^4*s+20*q^3*s^2-12*q^2*s^3+q*s^4+s^5-3*n^4-8*n^3*q+12*n^3*s-4*n ^2*q^2+24*n^2*q*s-18*n^2*s^2+4*n*q^3+8*n*q^2*s-24*n*q*s^2+12*n*s^3+3*q^4-4*q^3* s-4*q^2*s^2+8*q*s^3-3*s^4-2*n^3-10*n^2*q+6*n^2*s-11*n*q^2+20*n*q*s-6*n*s^2-3*q^ 3+11*q^2*s-10*q*s^2+2*s^3+2*n^2-2*n*q-4*n*s-3*q^2+2*q*s+2*s^2+3*n+q-3*s+1)*diff (diff(A[n+1](r,s),s),s)-2*(n^8+5*n^7*q-8*n^7*s+9*n^6*q^2-35*n^6*q*s+28*n^6*s^2+ 5*n^5*q^3-54*n^5*q^2*s+105*n^5*q*s^2-56*n^5*s^3-5*n^4*q^4-25*n^4*q^3*s+135*n^4* q^2*s^2-175*n^4*q*s^3+70*n^4*s^4-9*n^3*q^5+20*n^3*q^4*s+50*n^3*q^3*s^2-180*n^3* q^2*s^3+175*n^3*q*s^4-56*n^3*s^5-5*n^2*q^6+27*n^2*q^5*s-30*n^2*q^4*s^2-50*n^2*q ^3*s^3+135*n^2*q^2*s^4-105*n^2*q*s^5+28*n^2*s^6-n*q^7+10*n*q^6*s-27*n*q^5*s^2+ 20*n*q^4*s^3+25*n*q^3*s^4-54*n*q^2*s^5+35*n*q*s^6-8*n*s^7+q^7*s-5*q^6*s^2+9*q^5 *s^3-5*q^4*s^4-5*q^3*s^5+9*q^2*s^6-5*q*s^7+s^8+4*n^7+22*n^6*q-28*n^6*s+49*n^5*q ^2-132*n^5*q*s+84*n^5*s^2+55*n^4*q^3-245*n^4*q^2*s+330*n^4*q*s^2-140*n^4*s^3+30 *n^3*q^4-220*n^3*q^3*s+490*n^3*q^2*s^2-440*n^3*q*s^3+140*n^3*s^4+4*n^2*q^5-90*n ^2*q^4*s+330*n^2*q^3*s^2-490*n^2*q^2*s^3+330*n^2*q*s^4-84*n^2*s^5-3*n*q^6-8*n*q ^5*s+90*n*q^4*s^2-220*n*q^3*s^3+245*n*q^2*s^4-132*n*q*s^5+28*n*s^6-q^7+3*q^6*s+ 4*q^5*s^2-30*q^4*s^3+55*q^3*s^4-49*q^2*s^5+22*q*s^6-4*s^7+3*n^6+22*n^5*q-18*n^5 *s+60*n^4*q^2-110*n^4*q*s+45*n^4*s^2+80*n^3*q^3-240*n^3*q^2*s+220*n^3*q*s^2-60* n^3*s^3+55*n^2*q^4-240*n^2*q^3*s+360*n^2*q^2*s^2-220*n^2*q*s^3+45*n^2*s^4+18*n* q^5-110*n*q^4*s+240*n*q^3*s^2-240*n*q^2*s^3+110*n*q*s^4-18*n*s^5+2*q^6-18*q^5*s +55*q^4*s^2-80*q^3*s^3+60*q^2*s^4-22*q*s^5+3*s^6-7*n^5-19*n^4*q+35*n^4*s-10*n^3 *q^2+76*n^3*q*s-70*n^3*s^2+14*n^2*q^3+30*n^2*q^2*s-114*n^2*q*s^2+70*n^2*s^3+17* n*q^4-28*n*q^3*s-30*n*q^2*s^2+76*n*q*s^3-35*n*s^4+5*q^5-17*q^4*s+14*q^3*s^2+10* q^2*s^3-19*q*s^4+7*s^5-13*n^4-43*n^3*q+52*n^3*s-50*n^2*q^2+129*n^2*q*s-78*n^2*s ^2-23*n*q^3+100*n*q^2*s-129*n*q*s^2+52*n*s^3-3*q^4+23*q^3*s-50*q^2*s^2+43*q*s^3 -13*s^4-6*n^3-20*n^2*q+18*n^2*s-21*n*q^2+40*n*q*s-18*n*s^2-7*q^3+21*q^2*s-20*q* s^2+6*s^3+n^2-2*n*s-q^2+s^2+n+q-s)/(2*n^6*q+8*n^5*q^2-12*n^5*q*s+10*n^4*q^3-40* n^4*q^2*s+30*n^4*q*s^2-40*n^3*q^3*s+80*n^3*q^2*s^2-40*n^3*q*s^3-10*n^2*q^5+60*n ^2*q^3*s^2-80*n^2*q^2*s^3+30*n^2*q*s^4-8*n*q^6+20*n*q^5*s-40*n*q^3*s^3+40*n*q^2 *s^4-12*n*q*s^5-2*q^7+8*q^6*s-10*q^5*s^2+10*q^3*s^4-8*q^2*s^5+2*q*s^6-n^6-4*n^5 *q+6*n^5*s+n^4*q^2+20*n^4*q*s-15*n^4*s^2+24*n^3*q^3-4*n^3*q^2*s-40*n^3*q*s^2+20 *n^3*s^3+41*n^2*q^4-72*n^2*q^3*s+6*n^2*q^2*s^2+40*n^2*q*s^3-15*n^2*s^4+28*n*q^5 -82*n*q^4*s+72*n*q^3*s^2-4*n*q^2*s^3-20*n*q*s^4+6*n*s^5+7*q^6-28*q^5*s+41*q^4*s ^2-24*q^3*s^3+q^2*s^4+4*q*s^5-s^6-n^5-13*n^4*q+5*n^4*s-36*n^3*q^2+52*n^3*q*s-10 *n^3*s^2-40*n^2*q^3+108*n^2*q^2*s-78*n^2*q*s^2+10*n^2*s^3-19*n*q^4+80*n*q^3*s-\ 108*n*q^2*s^2+52*n*q*s^3-5*n*s^4-3*q^5+19*q^4*s-40*q^3*s^2+36*q^2*s^3-13*q*s^4+ s^5+4*n^4+4*n^3*q-16*n^3*s-13*n^2*q^2-12*n^2*q*s+24*n^2*s^2-22*n*q^3+26*n*q^2*s +12*n*q*s^2-16*n*s^3-9*q^4+22*q^3*s-13*q^2*s^2-4*q*s^3+4*s^4+6*n^3+20*n^2*q-18* n^2*s+17*n*q^2-40*n*q*s+18*n*s^2+3*q^3-17*q^2*s+20*q*s^2-6*s^3-n^2+8*n*q+2*n*s+ 7*q^2-8*q*s-s^2-5*n-q+5*s-2)/(-s-1+n+q)*diff(diff(diff(A[n+1](r,s),s),s),s)+(n^ 7+5*n^6*q-7*n^6*s+10*n^5*q^2-30*n^5*q*s+21*n^5*s^2+10*n^4*q^3-50*n^4*q^2*s+75*n ^4*q*s^2-35*n^4*s^3+5*n^3*q^4-40*n^3*q^3*s+100*n^3*q^2*s^2-100*n^3*q*s^3+35*n^3 *s^4+n^2*q^5-15*n^2*q^4*s+60*n^2*q^3*s^2-100*n^2*q^2*s^3+75*n^2*q*s^4-21*n^2*s^ 5-2*n*q^5*s+15*n*q^4*s^2-40*n*q^3*s^3+50*n*q^2*s^4-30*n*q*s^5+7*n*s^6+q^5*s^2-5 *q^4*s^3+10*q^3*s^4-10*q^2*s^5+5*q*s^6-s^7+5*n^6+22*n^5*q-30*n^5*s+38*n^4*q^2-\ 110*n^4*q*s+75*n^4*s^2+32*n^3*q^3-152*n^3*q^2*s+220*n^3*q*s^2-100*n^3*s^3+13*n^ 2*q^4-96*n^2*q^3*s+228*n^2*q^2*s^2-220*n^2*q*s^3+75*n^2*s^4+2*n*q^5-26*n*q^4*s+ 96*n*q^3*s^2-152*n*q^2*s^3+110*n*q*s^4-30*n*s^5-2*q^5*s+13*q^4*s^2-32*q^3*s^3+ 38*q^2*s^4-22*q*s^5+5*s^6+10*n^5+38*n^4*q-50*n^4*s+55*n^3*q^2-152*n^3*q*s+100*n ^3*s^2+37*n^2*q^3-165*n^2*q^2*s+228*n^2*q*s^2-100*n^2*s^3+11*n*q^4-74*n*q^3*s+ 165*n*q^2*s^2-152*n*q*s^3+50*n*s^4+q^5-11*q^4*s+37*q^3*s^2-55*q^2*s^3+38*q*s^4-\ 10*s^5+10*n^4+32*n^3*q-40*n^3*s+37*n^2*q^2-96*n^2*q*s+60*n^2*s^2+18*n*q^3-74*n* q^2*s+96*n*q*s^2-40*n*s^3+3*q^4-18*q^3*s+37*q^2*s^2-32*q*s^3+10*s^4+5*n^3+13*n^ 2*q-15*n^2*s+11*n*q^2-26*n*q*s+15*n*s^2+3*q^3-11*q^2*s+13*q*s^2-5*s^3+n^2+2*n*q -2*n*s+q^2-2*q*s+s^2)/(2*n^6*q+8*n^5*q^2-12*n^5*q*s+10*n^4*q^3-40*n^4*q^2*s+30* n^4*q*s^2-40*n^3*q^3*s+80*n^3*q^2*s^2-40*n^3*q*s^3-10*n^2*q^5+60*n^2*q^3*s^2-80 *n^2*q^2*s^3+30*n^2*q*s^4-8*n*q^6+20*n*q^5*s-40*n*q^3*s^3+40*n*q^2*s^4-12*n*q*s ^5-2*q^7+8*q^6*s-10*q^5*s^2+10*q^3*s^4-8*q^2*s^5+2*q*s^6-n^6-4*n^5*q+6*n^5*s+n^ 4*q^2+20*n^4*q*s-15*n^4*s^2+24*n^3*q^3-4*n^3*q^2*s-40*n^3*q*s^2+20*n^3*s^3+41*n ^2*q^4-72*n^2*q^3*s+6*n^2*q^2*s^2+40*n^2*q*s^3-15*n^2*s^4+28*n*q^5-82*n*q^4*s+ 72*n*q^3*s^2-4*n*q^2*s^3-20*n*q*s^4+6*n*s^5+7*q^6-28*q^5*s+41*q^4*s^2-24*q^3*s^ 3+q^2*s^4+4*q*s^5-s^6-n^5-13*n^4*q+5*n^4*s-36*n^3*q^2+52*n^3*q*s-10*n^3*s^2-40* n^2*q^3+108*n^2*q^2*s-78*n^2*q*s^2+10*n^2*s^3-19*n*q^4+80*n*q^3*s-108*n*q^2*s^2 +52*n*q*s^3-5*n*s^4-3*q^5+19*q^4*s-40*q^3*s^2+36*q^2*s^3-13*q*s^4+s^5+4*n^4+4*n ^3*q-16*n^3*s-13*n^2*q^2-12*n^2*q*s+24*n^2*s^2-22*n*q^3+26*n*q^2*s+12*n*q*s^2-\ 16*n*s^3-9*q^4+22*q^3*s-13*q^2*s^2-4*q*s^3+4*s^4+6*n^3+20*n^2*q-18*n^2*s+17*n*q ^2-40*n*q*s+18*n*s^2+3*q^3-17*q^2*s+20*q*s^2-6*s^3-n^2+8*n*q+2*n*s+7*q^2-8*q*s- s^2-5*n-q+5*s-2)*diff(diff(diff(diff(A[n+1](r,s),s),s),s),s) = 0 ------------------------------------------------- This took, 0.937, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ A[n](r, s) = ) / ----- k = 0 3 (k - 1 + p) (n - k + q) k binomial(n, k) binomial(n + k, k) (r + k) (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)^3*binomial(n+k,k)*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x^k ,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 3 3 3 2 3 2 3 2 3 3 2 2 4 (n p - 2 n p r + n p s + n p r - 2 n p r s + n r s - n p 2 3 2 3 2 2 2 2 2 2 2 3 2 + 2 n p r - n p s - n p r + 2 n p r s - n p r s - 6 n p 3 3 3 2 3 2 3 2 2 + 7 n p r - 5 n p s - n r + 5 n r s + 10 n p - 15 n p r 2 2 2 2 2 2 2 4 3 + 9 n p s + 5 n p r - 13 n p r s + 4 n r s - 2 n p + 4 n p r 3 2 2 2 2 3 3 - 2 n p s - 2 n p r + 4 n p r s - 2 n p r s + 11 n p - 5 n r 3 2 2 2 2 2 2 2 + 6 n s - 35 n p + 33 n p r - 26 n p s - 4 n r + 20 n r s 3 2 2 2 2 4 + 17 n p - 24 n p r + 15 n p s + 7 n p r - 20 n p r s + 5 n r s - p 3 3 2 2 2 2 3 2 2 + 2 p r - p s - p r + 2 p r s - p r s - 6 n + 50 n p - 20 n r 2 2 2 3 + 24 n s - 52 n p + 45 n p r - 37 n p s - 5 n r + 25 n r s + 8 p 2 2 2 2 2 - 11 p r + 7 p s + 3 p r - 9 p r s + 2 r s - 24 n + 67 n p - 25 n r 2 2 + 30 n s - 23 p + 19 p r - 16 p s - 2 r + 10 r s - 30 n + 28 p - 10 r / 2 2 + 12 s - 12) A[n](r, s) / ((n + 2 n r + r + 2 n + 2 r + 1) (n + r + 1) / 2 3 4 3 3 3 3 3 2 2 3 2 (p - r - 1) ) - (2 n p - 5 n p r + 3 n p s + 3 n p r - 9 n p r s 3 3 3 2 3 4 3 3 2 5 2 4 2 4 + n p r + 9 n p r s - n r - 3 n r s - n p + n p r - 2 n p s 2 3 2 2 3 2 2 3 2 2 2 2 4 + 3 n p r + 5 n p r s - 5 n p r - 3 n p r s + 2 n p r 2 3 2 4 3 3 3 2 3 2 3 2 - n p r s + n r s - 15 n p + 27 n p r - 18 n p s - 9 n p r 3 3 3 3 2 2 4 2 3 2 3 + 36 n p r s - 3 n r - 18 n r s + 15 n p - 21 n p r + 24 n p s 2 2 2 2 2 2 3 2 2 2 4 - 9 n p r - 54 n p r s + 21 n p r + 36 n p r s - 6 n r 2 3 5 4 4 3 2 3 - 6 n r s - 2 n p + 2 n p r - 4 n p s + 6 n p r + 10 n p r s 2 3 2 2 4 3 4 3 2 - 10 n p r - 6 n p r s + 4 n p r - 2 n p r s + 2 n r s + 40 n p 3 3 3 2 3 2 3 2 2 - 46 n p r + 34 n p s + 6 n r - 34 n r s - 76 n p + 94 n p r 2 2 2 2 2 2 3 2 2 4 - 94 n p s + 6 n p r + 154 n p r s - 24 n r - 60 n r s + 24 n p 3 3 2 2 2 3 - 27 n p r + 39 n p s - 27 n p r - 81 n p r s + 39 n p r 2 4 3 5 4 4 3 2 + 45 n p r s - 9 n r - 3 n r s - p + p r - 2 p s + 3 p r 3 2 3 2 2 4 3 4 3 + 5 p r s - 5 p r - 3 p r s + 2 p r - p r s + r s - 45 n p 3 3 2 2 2 2 2 + 24 n r - 21 n s + 171 n p - 150 n p r + 147 n p s - 126 n r s 3 2 2 2 3 - 107 n p + 107 n p r - 134 n p s + 39 n p r + 200 n p r s - 39 n r 2 4 3 3 2 2 2 3 - 66 n r s + 11 p - 11 p r + 18 p s - 15 p r - 36 p r s + 19 p r 2 4 3 2 2 2 2 + 18 p r s - 4 r + 18 n - 175 n p + 76 n r - 81 n s + 222 n p 2 3 2 2 - 162 n p r + 192 n p s - 18 n r - 150 n r s - 46 p + 40 p r - 58 p s 2 3 2 2 + 24 p r + 82 p r s - 18 r - 24 r s + 66 n - 215 n p + 80 n r - 99 n s 2 2 + 91 p - 58 p r + 79 p s - 12 r - 58 r s + 78 n - 85 p + 28 r - 39 s /d \ / 2 3 3 + 30) |-- A[n](r, s)| / ((p - r - 1) %1 (p - r - 2)) + (n p \dr / / 3 2 3 2 3 3 3 3 2 2 3 + 3 n p s - 3 n p r - 6 n p r s + 2 n r + 3 n r s - 2 n p r 2 3 2 2 2 2 2 2 4 2 3 3 2 - n p s + 3 n p r + 3 n p r s - n r - 2 n r s - 6 n p 3 3 2 3 2 3 2 2 2 2 - 12 n p s + 6 n r + 12 n r s + 3 n p + 12 n p r + 15 n p s 2 2 2 2 3 2 2 3 3 - 21 n p r - 18 n p r s + 6 n r + 3 n r s - 4 n p r - 2 n p s 2 2 2 4 3 3 3 + 6 n p r + 6 n p r s - 2 n r - 4 n r s + 11 n p + 11 n s 2 2 2 2 2 2 2 3 - 18 n p - 22 n p r - 47 n p s + 29 n r + 36 n r s + 3 n p 2 2 2 3 2 + 24 n p r + 21 n p s - 33 n p r - 18 n p r s + 6 n r - 3 n r s 3 3 2 2 2 4 3 3 2 - 2 p r - p s + 3 p r + 3 p r s - r - 2 r s - 6 n + 33 n p 2 2 2 2 + 12 n r + 39 n s - 18 n p - 44 n p r - 58 n p s + 40 n r + 36 n r s 3 2 2 2 3 2 2 + p + 12 p r + 9 p s - 15 p r - 6 p r s + 2 r - 3 r s - 18 n 2 2 + 33 n p + 24 n r + 45 n s - 6 p - 22 p r - 23 p s + 17 r + 12 r s / 2 \ |d | / - 18 n + 11 p + 12 r + 17 s - 6) |--- A[n](r, s)| / ( | 2 | / \dr / 2 2 (p - 2 p r + r - 3 p + 3 r + 2) (n + r + 1) 2 2 3 3 2 2 2 2 (n + 2 n r + r + 2 n + 2 r + 1)) - (n r + n s - n r - n r s + 3 n r 2 2 2 + 3 n s - 2 n r - 2 n r s + 3 n r + 3 n s - r - r s + r + s) / 3 \ |d | 3 2 3 3 2 2 3 2 2 |--- A[n](r, s)|/(%1) - (n p - 2 n p r + n r + 3 n p - 6 n p r | 3 | \dr / 2 2 4 3 2 2 5 4 3 2 3 + 3 n p r + 3 n p - 6 n p r + 3 n p r + p - 2 p r + p r - 5 n p 3 2 2 2 3 2 4 3 + 5 n r - 15 n p + 15 n p r - 15 n p + 15 n p r - 5 p + 5 p r 3 2 2 3 / 2 + 6 n + 18 n p + 18 n p + 6 p ) A[n + 1](r, s) / ((p - r - 1) %1) + / 2 3 2 2 2 2 2 3 4 3 (3 n p - 9 n p r + 9 n p r - 3 n r + 6 n p - 18 n p r 2 2 3 5 4 3 2 2 3 2 2 + 18 n p r - 6 n p r + 3 p - 9 p r + 9 p r - 3 p r - 18 n p 2 2 2 3 2 2 3 4 + 36 n p r - 18 n r - 39 n p + 81 n p r - 45 n p r + 3 n r - 21 p 3 2 2 3 2 2 2 + 45 p r - 27 p r + 3 p r + 34 n p - 34 n r + 84 n p - 100 n p r 2 3 2 2 3 2 + 16 n r + 51 p - 69 p r + 19 p r - r - 21 n - 67 n p + 25 n r 2 2 /d \ / - 51 p + 35 p r - 5 r + 12 n + 19 p - 7 r - 3) |-- A[n + 1](r, s)| / ( \dr / / 2 2 (p - r - 1) (n + 2 n r + r + 2 n + 2 r + 1) 2 2 2 2 3 (p - 2 p r + r - 3 p + 3 r + 2)) - (3 n p - 6 n p r + 3 n r + 3 p 2 2 2 2 - 6 p r + 3 p r - 12 n p + 12 n r - 15 p + 18 p r - 3 r + 11 n + 21 p / 2 \ |d | / 2 2 - 10 r - 7) |--- A[n + 1](r, s)| / ((p - 2 p r + r - 3 p + 3 r + 2) | 2 | / \dr / / 3 \ |d | (n + r + 1)) + |--- A[n + 1](r, s)| = 0 | 3 | \dr / 3 2 2 3 2 2 %1 := n + 3 n r + 3 n r + r + 3 n + 6 n r + 3 r + 3 n + 3 r + 1 5 4 4 3 2 3 3 2 2 3 2 2 - (2 n + 5 n q - 4 n s + 4 n q - 6 n q s + 2 n s + n q - 2 n q s 2 2 4 3 3 2 2 2 2 2 + n q s + 5 n + 12 n q - 10 n s + 9 n q - 14 n q s + 5 n s 3 2 2 3 2 2 2 + 2 n q - 4 n q s + 2 n q s + 2 n + 8 n q - 8 n s + 6 n q 2 3 2 2 2 2 2 - 10 n q s + 4 n s + q - 2 q s + q s - 4 n - 2 n s + q - 2 q s + s / - 4 n - q - 1) A[n](r, s) / ( / 3 2 2 3 2 2 (n - 3 n s + 3 n s - s + 3 n - 6 n s + 3 s + 3 n - 3 s + 1) 5 4 4 3 2 3 3 2 2 3 (n + q - s)) + (3 n + 7 n q - 5 n s + 5 n q - 6 n q s + n s + n q 2 2 2 2 2 3 4 3 3 2 2 - n q s - n q s + n s + 9 n + 19 n q - 14 n s + 12 n q 2 2 2 3 2 2 3 3 - 15 n q s + 3 n s + 2 n q - 2 n q s - 2 n q s + 2 n s + 8 n 2 2 2 2 3 2 2 3 + 17 n q - 14 n s + 9 n q - 12 n q s + 3 n s + q - q s - q s + s 2 2 /d \ / + 5 n q - 6 n s + 2 q - 3 q s + s - 3 n - s - 1) |-- A[n](r, s)| / ( \ds / / 3 4 3 2 2 2 3 2 (n + q - s) (n - s + 1) ) - (n + n q + n q s - n s + 4 n + 3 n q 2 2 2 + 2 n q s - 2 n s + 6 n + 3 n q + q s - s + 4 n + q + 1) / 2 \ |d | / |--- A[n](r, s)| / ( | 2 | / \ds / 3 2 2 3 2 2 n - 3 n s + 3 n s - s + 3 n - 6 n s + 3 s + 3 n - 3 s + 1) 2 2 2 3 (n + 2 n q - 2 n s + q - 2 q s + s - n - q + s) q A[n + 1](r, s) + -------------------------------------------------------------------- + ( 2 3 (n + q - s + 1) (n - s + 1) 3 2 2 3 2 2 4 3 2 2 5 3 n q + 9 n q - 9 n q s + 9 n q - 18 n q s + 9 n q s + 3 q 4 3 2 2 3 3 2 2 2 3 - 9 q s + 9 q s - 3 q s - 3 n q - 9 n q + 9 n q s - 9 n q 2 2 4 3 2 2 3 3 2 + 18 n q s - 9 n q s - 3 q + 9 q s - 9 q s + 3 q s + n + n q 2 2 2 3 2 2 3 2 - 3 n s - 3 n q - 2 n q s + 3 n s - 3 q + 3 q s + q s - s + n 2 2 /d \ + 5 n q - 2 n s + 3 q - 5 q s + s - n + q + s - 1) |-- A[n + 1](r, s)| \ds / / 2 / ((n + q - s + 1) (n - s + 1) / 2 2 2 2 2 (n + 2 n q - 2 n s + q - 2 q s + s + n + q - s)) + (3 n q + 6 n q 3 2 2 2 2 - 6 n q s + 3 q - 6 q s + 3 q s - 3 n - 6 n q + 6 n s - 3 q + 6 q s / 2 \ 2 |d | / - 3 s - 2 n - 3 q + 2 s + 1) |--- A[n + 1](r, s)| / ( | 2 | / \ds / 2 2 2 (n + 2 n q - 2 n s + q - 2 q s + s + n + q - s) (n - s + 1)) / 3 \ |d | + |--- A[n + 1](r, s)| = 0 | 3 | \ds / and in Maple notation (n^3*p^3-2*n^3*p^2*r+n^3*p^2*s+n^3*p*r^2-2*n^3*p*r*s+n^3*r^2*s-n^2*p^4+2*n^2*p^ 3*r-n^2*p^3*s-n^2*p^2*r^2+2*n^2*p^2*r*s-n^2*p*r^2*s-6*n^3*p^2+7*n^3*p*r-5*n^3*p *s-n^3*r^2+5*n^3*r*s+10*n^2*p^3-15*n^2*p^2*r+9*n^2*p^2*s+5*n^2*p*r^2-13*n^2*p*r *s+4*n^2*r^2*s-2*n*p^4+4*n*p^3*r-2*n*p^3*s-2*n*p^2*r^2+4*n*p^2*r*s-2*n*p*r^2*s+ 11*n^3*p-5*n^3*r+6*n^3*s-35*n^2*p^2+33*n^2*p*r-26*n^2*p*s-4*n^2*r^2+20*n^2*r*s+ 17*n*p^3-24*n*p^2*r+15*n*p^2*s+7*n*p*r^2-20*n*p*r*s+5*n*r^2*s-p^4+2*p^3*r-p^3*s -p^2*r^2+2*p^2*r*s-p*r^2*s-6*n^3+50*n^2*p-20*n^2*r+24*n^2*s-52*n*p^2+45*n*p*r-\ 37*n*p*s-5*n*r^2+25*n*r*s+8*p^3-11*p^2*r+7*p^2*s+3*p*r^2-9*p*r*s+2*r^2*s-24*n^2 +67*n*p-25*n*r+30*n*s-23*p^2+19*p*r-16*p*s-2*r^2+10*r*s-30*n+28*p-10*r+12*s-12) /(n^2+2*n*r+r^2+2*n+2*r+1)/(n+r+1)/(p-r-1)^2*A[n](r,s)-(2*n^3*p^4-5*n^3*p^3*r+3 *n^3*p^3*s+3*n^3*p^2*r^2-9*n^3*p^2*r*s+n^3*p*r^3+9*n^3*p*r^2*s-n^3*r^4-3*n^3*r^ 3*s-n^2*p^5+n^2*p^4*r-2*n^2*p^4*s+3*n^2*p^3*r^2+5*n^2*p^3*r*s-5*n^2*p^2*r^3-3*n ^2*p^2*r^2*s+2*n^2*p*r^4-n^2*p*r^3*s+n^2*r^4*s-15*n^3*p^3+27*n^3*p^2*r-18*n^3*p ^2*s-9*n^3*p*r^2+36*n^3*p*r*s-3*n^3*r^3-18*n^3*r^2*s+15*n^2*p^4-21*n^2*p^3*r+24 *n^2*p^3*s-9*n^2*p^2*r^2-54*n^2*p^2*r*s+21*n^2*p*r^3+36*n^2*p*r^2*s-6*n^2*r^4-6 *n^2*r^3*s-2*n*p^5+2*n*p^4*r-4*n*p^4*s+6*n*p^3*r^2+10*n*p^3*r*s-10*n*p^2*r^3-6* n*p^2*r^2*s+4*n*p*r^4-2*n*p*r^3*s+2*n*r^4*s+40*n^3*p^2-46*n^3*p*r+34*n^3*p*s+6* n^3*r^2-34*n^3*r*s-76*n^2*p^3+94*n^2*p^2*r-94*n^2*p^2*s+6*n^2*p*r^2+154*n^2*p*r *s-24*n^2*r^3-60*n^2*r^2*s+24*n*p^4-27*n*p^3*r+39*n*p^3*s-27*n*p^2*r^2-81*n*p^2 *r*s+39*n*p*r^3+45*n*p*r^2*s-9*n*r^4-3*n*r^3*s-p^5+p^4*r-2*p^4*s+3*p^3*r^2+5*p^ 3*r*s-5*p^2*r^3-3*p^2*r^2*s+2*p*r^4-p*r^3*s+r^4*s-45*n^3*p+24*n^3*r-21*n^3*s+ 171*n^2*p^2-150*n^2*p*r+147*n^2*p*s-126*n^2*r*s-107*n*p^3+107*n*p^2*r-134*n*p^2 *s+39*n*p*r^2+200*n*p*r*s-39*n*r^3-66*n*r^2*s+11*p^4-11*p^3*r+18*p^3*s-15*p^2*r ^2-36*p^2*r*s+19*p*r^3+18*p*r^2*s-4*r^4+18*n^3-175*n^2*p+76*n^2*r-81*n^2*s+222* n*p^2-162*n*p*r+192*n*p*s-18*n*r^2-150*n*r*s-46*p^3+40*p^2*r-58*p^2*s+24*p*r^2+ 82*p*r*s-18*r^3-24*r^2*s+66*n^2-215*n*p+80*n*r-99*n*s+91*p^2-58*p*r+79*p*s-12*r ^2-58*r*s+78*n-85*p+28*r-39*s+30)/(p-r-1)^2/(n^3+3*n^2*r+3*n*r^2+r^3+3*n^2+6*n* r+3*r^2+3*n+3*r+1)/(p-r-2)*diff(A[n](r,s),r)+(n^3*p^3+3*n^3*p^2*s-3*n^3*p*r^2-6 *n^3*p*r*s+2*n^3*r^3+3*n^3*r^2*s-2*n^2*p^3*r-n^2*p^3*s+3*n^2*p^2*r^2+3*n^2*p*r^ 2*s-n^2*r^4-2*n^2*r^3*s-6*n^3*p^2-12*n^3*p*s+6*n^3*r^2+12*n^3*r*s+3*n^2*p^3+12* n^2*p^2*r+15*n^2*p^2*s-21*n^2*p*r^2-18*n^2*p*r*s+6*n^2*r^3+3*n^2*r^2*s-4*n*p^3* r-2*n*p^3*s+6*n*p^2*r^2+6*n*p*r^2*s-2*n*r^4-4*n*r^3*s+11*n^3*p+11*n^3*s-18*n^2* p^2-22*n^2*p*r-47*n^2*p*s+29*n^2*r^2+36*n^2*r*s+3*n*p^3+24*n*p^2*r+21*n*p^2*s-\ 33*n*p*r^2-18*n*p*r*s+6*n*r^3-3*n*r^2*s-2*p^3*r-p^3*s+3*p^2*r^2+3*p*r^2*s-r^4-2 *r^3*s-6*n^3+33*n^2*p+12*n^2*r+39*n^2*s-18*n*p^2-44*n*p*r-58*n*p*s+40*n*r^2+36* n*r*s+p^3+12*p^2*r+9*p^2*s-15*p*r^2-6*p*r*s+2*r^3-3*r^2*s-18*n^2+33*n*p+24*n*r+ 45*n*s-6*p^2-22*p*r-23*p*s+17*r^2+12*r*s-18*n+11*p+12*r+17*s-6)/(p^2-2*p*r+r^2-\ 3*p+3*r+2)/(n+r+1)/(n^2+2*n*r+r^2+2*n+2*r+1)*diff(diff(A[n](r,s),r),r)-(n^3*r+n ^3*s-n^2*r^2-n^2*r*s+3*n^2*r+3*n^2*s-2*n*r^2-2*n*r*s+3*n*r+3*n*s-r^2-r*s+r+s)/( n^3+3*n^2*r+3*n*r^2+r^3+3*n^2+6*n*r+3*r^2+3*n+3*r+1)*diff(diff(diff(A[n](r,s),r ),r),r)-(n^3*p^2-2*n^3*p*r+n^3*r^2+3*n^2*p^3-6*n^2*p^2*r+3*n^2*p*r^2+3*n*p^4-6* n*p^3*r+3*n*p^2*r^2+p^5-2*p^4*r+p^3*r^2-5*n^3*p+5*n^3*r-15*n^2*p^2+15*n^2*p*r-\ 15*n*p^3+15*n*p^2*r-5*p^4+5*p^3*r+6*n^3+18*n^2*p+18*n*p^2+6*p^3)/(p-r-1)^2/(n^3 +3*n^2*r+3*n*r^2+r^3+3*n^2+6*n*r+3*r^2+3*n+3*r+1)*A[n+1](r,s)+(3*n^2*p^3-9*n^2* p^2*r+9*n^2*p*r^2-3*n^2*r^3+6*n*p^4-18*n*p^3*r+18*n*p^2*r^2-6*n*p*r^3+3*p^5-9*p ^4*r+9*p^3*r^2-3*p^2*r^3-18*n^2*p^2+36*n^2*p*r-18*n^2*r^2-39*n*p^3+81*n*p^2*r-\ 45*n*p*r^2+3*n*r^3-21*p^4+45*p^3*r-27*p^2*r^2+3*p*r^3+34*n^2*p-34*n^2*r+84*n*p^ 2-100*n*p*r+16*n*r^2+51*p^3-69*p^2*r+19*p*r^2-r^3-21*n^2-67*n*p+25*n*r-51*p^2+ 35*p*r-5*r^2+12*n+19*p-7*r-3)/(p-r-1)/(n^2+2*n*r+r^2+2*n+2*r+1)/(p^2-2*p*r+r^2-\ 3*p+3*r+2)*diff(A[n+1](r,s),r)-(3*n*p^2-6*n*p*r+3*n*r^2+3*p^3-6*p^2*r+3*p*r^2-\ 12*n*p+12*n*r-15*p^2+18*p*r-3*r^2+11*n+21*p-10*r-7)/(p^2-2*p*r+r^2-3*p+3*r+2)/( n+r+1)*diff(diff(A[n+1](r,s),r),r)+diff(diff(diff(A[n+1](r,s),r),r),r) = 0 -(2*n^5+5*n^4*q-4*n^4*s+4*n^3*q^2-6*n^3*q*s+2*n^3*s^2+n^2*q^3-2*n^2*q^2*s+n^2*q *s^2+5*n^4+12*n^3*q-10*n^3*s+9*n^2*q^2-14*n^2*q*s+5*n^2*s^2+2*n*q^3-4*n*q^2*s+2 *n*q*s^2+2*n^3+8*n^2*q-8*n^2*s+6*n*q^2-10*n*q*s+4*n*s^2+q^3-2*q^2*s+q*s^2-4*n^2 -2*n*s+q^2-2*q*s+s^2-4*n-q-1)/(n^3-3*n^2*s+3*n*s^2-s^3+3*n^2-6*n*s+3*s^2+3*n-3* s+1)/(n+q-s)*A[n](r,s)+(3*n^5+7*n^4*q-5*n^4*s+5*n^3*q^2-6*n^3*q*s+n^3*s^2+n^2*q ^3-n^2*q^2*s-n^2*q*s^2+n^2*s^3+9*n^4+19*n^3*q-14*n^3*s+12*n^2*q^2-15*n^2*q*s+3* n^2*s^2+2*n*q^3-2*n*q^2*s-2*n*q*s^2+2*n*s^3+8*n^3+17*n^2*q-14*n^2*s+9*n*q^2-12* n*q*s+3*n*s^2+q^3-q^2*s-q*s^2+s^3+5*n*q-6*n*s+2*q^2-3*q*s+s^2-3*n-s-1)/(n+q-s)/ (n-s+1)^3*diff(A[n](r,s),s)-(n^4+n^3*q+n^2*q*s-n^2*s^2+4*n^3+3*n^2*q+2*n*q*s-2* n*s^2+6*n^2+3*n*q+q*s-s^2+4*n+q+1)/(n^3-3*n^2*s+3*n*s^2-s^3+3*n^2-6*n*s+3*s^2+3 *n-3*s+1)*diff(diff(A[n](r,s),s),s)+(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2-n-q+s)*q^3/( n+q-s+1)^2/(n-s+1)^3*A[n+1](r,s)+(3*n^3*q^2+9*n^2*q^3-9*n^2*q^2*s+9*n*q^4-18*n* q^3*s+9*n*q^2*s^2+3*q^5-9*q^4*s+9*q^3*s^2-3*q^2*s^3-3*n^3*q-9*n^2*q^2+9*n^2*q*s -9*n*q^3+18*n*q^2*s-9*n*q*s^2-3*q^4+9*q^3*s-9*q^2*s^2+3*q*s^3+n^3+n^2*q-3*n^2*s -3*n*q^2-2*n*q*s+3*n*s^2-3*q^3+3*q^2*s+q*s^2-s^3+n^2+5*n*q-2*n*s+3*q^2-5*q*s+s^ 2-n+q+s-1)/(n+q-s+1)/(n-s+1)^2/(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2+n+q-s)*diff(A[n+1 ](r,s),s)+(3*n^2*q+6*n*q^2-6*n*q*s+3*q^3-6*q^2*s+3*q*s^2-3*n^2-6*n*q+6*n*s-3*q^ 2+6*q*s-3*s^2-2*n-3*q+2*s+1)/(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2+n+q-s)/(n-s+1)*diff (diff(A[n+1](r,s),s),s)+diff(diff(diff(A[n+1](r,s),s),s),s) = 0 ------------------------------------------------- This took, 0.258, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ 3 2 (k - 1 + p) A[n](r, s) = ) binomial(n, k) binomial(n + k, k) (r + k) / ----- k = 0 (n - k + q) k (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)^3*binomial(n+k,k)^2*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x ^k,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 3 3 2 2 2 3 2 3 2 2 - (n p + n s - 2 n p - 2 n p s + n p + n p s - n + 7 n p + 5 n s 2 3 2 2 2 - 7 n p - 6 n p s + p + p s - 5 n + 14 n p + 8 n s - 5 p - 4 p s / 3 2 2 3 - 8 n + 8 p + 4 s - 4) A[n](r, s) / (n + 3 n p + 3 n p + p ) + ( / 3 3 3 2 3 2 3 3 3 3 2 2 4 2 n p - 3 n p r + 3 n p s - 6 n p r s + n r + 3 n r s - 2 n p 2 3 2 2 2 2 2 2 3 2 3 4 - 4 n p s + 6 n p r + 6 n p r s - 4 n p r - 2 n r s + 3 n p r 4 3 2 2 3 2 2 3 3 2 + n p s - 6 n p r + 3 n p r - 3 n p r s + 2 n p r s - 9 n p 3 3 3 2 3 2 2 2 2 + 9 n p r - 9 n p s + 9 n r s + 18 n p - 9 n p r + 27 n p s 2 2 2 2 3 2 2 4 3 - 18 n p r - 36 n p r s + 9 n r + 9 n r s - 4 n p - 18 n p r 3 2 2 2 3 2 3 - 14 n p s + 39 n p r + 12 n p r s - 17 n p r + 9 n p r s - 7 n r s 4 4 3 2 2 3 2 2 3 3 + 3 p r + p s - 6 p r + 3 p r - 3 p r s + 2 p r s + 13 n p 3 3 2 2 2 2 2 2 - 6 n r + 7 n s - 53 n p + 27 n p r - 53 n p s + 12 n r 2 3 2 2 2 + 39 n r s + 30 n p + 30 n p r + 58 n p s - 75 n p r - 54 n p r s 3 2 4 3 3 2 2 2 + 22 n r + 3 n r s - 2 p - 18 p r - 10 p s + 33 p r + 6 p r s 3 2 3 3 2 2 2 - 13 p r + 9 p r s - 5 r s - 6 n + 63 n p - 18 n r + 33 n s 2 2 3 2 - 79 n p - 9 n p r - 91 n p s + 42 n r + 51 n r s + 14 p + 36 p r 2 2 3 2 2 + 34 p s - 57 p r - 24 p r s + 14 r - 3 r s - 26 n + 87 n p - 6 n r 2 2 + 49 n s - 35 p - 27 p r - 47 p s + 30 r + 21 r s - 34 n + 37 p + 6 r /d \ 3 4 3 3 + 23 s - 14) |-- A[n](r, s)|/(%1 (p - r - 2)) - (n p - n p r \dr / 3 3 3 2 2 3 2 3 3 3 2 3 4 + 3 n p s - 3 n p r - 9 n p r s + 5 n p r + 9 n p r s - 2 n r 3 3 2 4 2 4 2 3 2 2 3 - 3 n r s - 4 n p r - 2 n p s + 10 n p r + 2 n p r s 2 2 3 2 2 2 2 4 2 3 2 5 - 6 n p r + 6 n p r s - 2 n p r - 10 n p r s + 2 n r 2 4 4 2 4 3 3 3 2 2 4 + 4 n r s + 3 n p r + 2 n p r s - 9 n p r - 5 n p r s + 9 n p r 2 3 5 4 5 3 3 3 2 + 3 n p r s - 3 n p r + n p r s - n r s - 7 n p + 6 n p r 3 2 3 2 3 3 3 3 2 2 4 - 15 n p s + 9 n p r + 30 n p r s - 8 n r - 15 n r s + 3 n p 2 3 2 3 2 2 2 2 2 2 3 + 25 n p r + 23 n p s - 63 n p r - 39 n p r s + 39 n p r 2 2 2 4 2 3 4 4 3 2 + 9 n p r s - 4 n r + 7 n r s - 8 n p r - 4 n p s - n p r 3 2 3 2 2 4 3 - 10 n p r s + 36 n p r + 39 n p r s - 37 n p r - 32 n p r s 5 4 4 2 4 3 3 3 2 2 4 + 10 n r + 7 n r s + 3 p r + 2 p r s - 9 p r - 5 p r s + 9 p r 2 3 5 4 5 3 2 3 3 + 3 p r s - 3 p r + p r s - r s + 17 n p - 11 n p r + 23 n p s 3 2 3 2 3 2 2 2 2 2 2 - 6 n r - 23 n r s - 21 n p - 50 n p r - 79 n p s + 117 n p r 2 2 3 2 2 4 3 3 + 112 n p r s - 46 n r - 33 n r s + 3 n p + 53 n p r + 37 n p s 2 2 2 3 2 4 3 - 66 n p r - 17 n p r s - 16 n p r - 54 n p r s + 26 n r + 34 n r s 4 4 3 2 3 2 3 2 2 4 - 4 p r - 2 p s - 11 p r - 12 p r s + 42 p r + 33 p r s - 35 p r 3 5 4 3 3 3 2 2 - 22 p r s + 8 r + 3 r s - 17 n p + 6 n r - 11 n s + 51 n p 2 2 2 2 2 3 2 + 35 n p r + 103 n p s - 64 n r - 81 n r s - 21 n p - 118 n p r 2 2 3 2 4 - 113 n p s + 156 n p r + 100 n p r s - 28 n r + 2 n r s + p 3 3 2 2 2 3 2 4 + 27 p r + 17 p s - 6 p r + 13 p r s - 50 p r - 54 p r s + 28 r 3 3 2 2 2 2 + 24 r s + 6 n - 51 n p - 6 n r - 45 n s + 51 n p + 103 n p r 2 3 2 2 2 + 137 n p s - 92 n r - 81 n r s - 7 p - 62 p r - 49 p s + 48 p r 3 2 2 2 + 18 p r s + 10 r + 20 r s + 18 n - 51 n p - 30 n r - 57 n s + 17 p 2 + 57 p r + 57 p s - 34 r - 23 r s + 18 n - 17 p - 18 r - 23 s + 6) / 2 \ |d | / |--- A[n](r, s)| / ( | 2 | / \dr / 3 2 2 3 2 2 (p - 3 p r + 3 p r - r - 7 p + 14 p r - 7 r + 16 p - 16 r - 12) 3 2 2 3 3 2 3 2 3 2 3 (n + 3 n p + 3 n p + p )) + (n p r + n p s - 2 n p r - 2 n p r s 3 3 3 2 2 2 2 2 2 2 3 2 2 + n r + n r s - 2 n p r - 2 n p r s + 4 n p r + 4 n p r s 2 4 2 3 2 3 2 2 4 3 5 - 2 n r - 2 n r s + n p r + n p r s - 2 n p r - 2 n p r s + n r 4 3 3 3 2 3 2 2 + n r s - 2 n p r - 2 n p s + 2 n r + 2 n r s + 3 n p r 2 2 2 2 2 2 3 2 2 2 2 + 3 n p s - 2 n p r - 2 n p r s - n r - n r s - 4 n p r 2 3 2 4 3 2 3 2 2 - 4 n p r s + 6 n p r + 6 n p r s - 2 n r - 2 n r s + p r + p r s 4 3 5 4 3 3 2 2 - 2 p r - 2 p r s + r + r s + n r + n s - 6 n p r - 6 n p s 2 2 2 2 2 2 3 + 4 n r + 4 n r s + 3 n p r + 3 n p s + 2 n p r + 2 n p r s - 4 n r 2 2 2 2 3 2 2 2 - 4 n r s - 2 p r - 2 p r s + 2 p r + 2 p r s + 3 n r + 3 n s 2 2 2 2 - 6 n p r - 6 n p s + 2 n r + 2 n r s + p r + p s + 2 p r + 2 p r s / 3 \ 3 2 |d | - 2 r - 2 r s + 3 n r + 3 n s - 2 p r - 2 p s + r + s) |--- A[n](r, s)|/ | 3 | \dr / 3 2 3 3 2 (%1 (-3 + p - r)) + A[n + 1](r, s) - (3 n p - 6 n p r + 3 n r 2 3 2 2 2 3 4 2 2 3 4 + 6 n p - 9 n p r + 3 n r + 3 n p - 9 n p r + 6 n p r + 3 p r 3 2 2 3 3 3 2 2 2 2 2 - 6 p r + 3 p r - 9 n p + 9 n r - 18 n p + 9 n p r + 9 n r 3 2 2 3 4 3 2 2 - 6 n p - 18 n p r + 27 n p r - 3 n r + 3 p - 18 p r + 18 p r 3 3 2 2 2 2 3 - 3 p r + 7 n + 12 n p + 9 n r - 6 n p + 36 n p r - 9 n r - 12 p 2 2 3 2 2 2 + 30 p r - 12 p r + r + 3 n + 15 n p - 9 n r + 15 p - 15 p r + 3 r /d \ / 3 2 3 3 2 - 3 n - 6 p + 3 r + 1) |-- A[n + 1](r, s)| / (n p - 2 n p r + n r \dr / / 2 3 2 2 2 2 4 3 2 2 5 + 3 n p - 6 n p r + 3 n p r + 3 n p - 6 n p r + 3 n p r + p 4 3 2 3 3 2 2 2 3 - 2 p r + p r - 4 n p + 4 n r - 12 n p + 12 n p r - 12 n p 2 4 3 3 2 2 3 3 3 + 12 n p r - 4 p + 4 p r + 4 n + 12 n p + 12 n p + 4 p ) + (3 n p 3 2 3 2 3 3 2 4 2 3 2 2 2 - 9 n p r + 9 n p r - 3 n r + 3 n p - 3 n p r - 9 n p r 2 3 2 4 4 3 2 2 3 4 + 15 n p r - 6 n r + 6 n p r - 15 n p r + 9 n p r + 3 n p r 5 4 2 3 3 2 4 5 3 2 3 - 3 n r + 3 p r - 9 p r + 9 p r - 3 p r - 15 n p + 30 n p r 3 2 2 3 2 2 2 2 2 3 4 - 15 n r - 12 n p - 9 n p r + 54 n p r - 33 n r + 6 n p 3 2 2 3 4 4 3 2 - 48 n p r + 63 n p r - 6 n p r - 15 n r + 6 p r - 36 p r 2 3 4 5 3 3 2 2 2 + 57 p r - 30 p r + 3 r + 23 n p - 23 n r + 6 n p + 57 n p r 2 2 3 2 2 3 4 3 - 63 n r - 33 n p + 111 n p r - 54 n p r - 24 n r + 3 p - 45 p r 2 2 3 4 3 2 2 2 + 123 p r - 100 p r + 19 r - 11 n + 18 n p - 51 n r + 57 n p 2 3 2 2 3 2 - 78 n p r - 12 n r - 18 p + 111 p r - 150 p r + 46 r - 15 n 2 2 - 33 n p + 3 n r + 36 p - 105 p r + 54 r + 3 n - 28 p + 31 r + 7) / 2 \ |d | / 3 2 3 3 2 2 3 |--- A[n + 1](r, s)| / ((-3 + p - r) (n p - 2 n p r + n r + 3 n p | 2 | / \dr / 2 2 2 2 4 3 2 2 5 4 - 6 n p r + 3 n p r + 3 n p - 6 n p r + 3 n p r + p - 2 p r 3 2 3 3 2 2 2 3 2 + p r - 4 n p + 4 n r - 12 n p + 12 n p r - 12 n p + 12 n p r 4 3 3 2 2 3 3 2 3 - 4 p + 4 p r + 4 n + 12 n p + 12 n p + 4 p )) - (n p - 2 n p r 3 2 2 2 2 2 2 3 2 2 3 4 + n r + 3 n p r - 6 n p r + 3 n r + 3 n p r - 6 n p r + 3 n r 2 3 4 5 3 3 2 2 2 2 2 + p r - 2 p r + r - 2 n p + 2 n r + 3 n p - 12 n p r + 9 n r 2 2 3 2 2 3 4 3 2 + 6 n p r - 18 n p r + 12 n r + 3 p r - 8 p r + 5 r + n - 6 n p 2 2 2 2 2 3 2 + 9 n r + 3 n p - 18 n p r + 18 n r + 3 p r - 12 p r + 10 r + 3 n 2 2 - 6 n p + 12 n r + p - 8 p r + 10 r + 3 n - 2 p + 5 r + 1) / 3 \ |d | |--- A[n + 1](r, s)|/(%1 (-3 + p - r)) = 0 | 3 | \dr / 3 3 2 2 2 3 2 4 3 3 %1 := n p - n r + 3 n p - 3 n p r + 3 n p - 3 n p r + p - p r - 2 n 2 2 3 - 6 n p - 6 n p - 2 p 5 4 4 3 2 3 3 2 2 3 - (4 n + 12 n q - 8 n s + 13 n q - 16 n q s + 4 n s + 6 n q 2 2 2 2 4 3 2 2 4 3 - 10 n q s + 4 n q s + n q - 2 n q s + n q s + 8 n + 22 n q 3 2 2 2 2 2 3 2 2 - 16 n s + 21 n q - 28 n q s + 8 n s + 8 n q - 14 n q s + 6 n q s 4 3 2 2 3 2 2 2 2 + q - 2 q s + q s + n + 8 n q - 10 n s + 8 n q - 14 n q s + 5 n s 3 2 2 2 2 + 2 q - 4 q s + 2 q s - 7 n - 4 n q - 2 n s - 2 q s + s - 5 n - 2 q / 2 2 - 1) A[n](r, s) / ((n - s + 1) (n - 2 n s + s + 2 n - 2 s + 1) / 5 4 4 3 2 3 3 2 (n + q - s)) + (4 n + 10 n q - 4 n s + 8 n q - 2 n q s - 4 n s 2 3 2 2 2 2 2 3 3 2 2 + 2 n q + 4 n q s - 10 n q s + 4 n s + 2 n q s - 4 n q s 3 4 3 3 2 2 2 2 2 + 2 n q s + 11 n + 25 n q - 13 n s + 18 n q - 12 n q s - 3 n s 3 2 2 3 3 2 2 3 + 4 n q + 2 n q s - 11 n q s + 5 n s + 2 q s - 4 q s + 2 q s 3 2 2 2 2 3 2 + 8 n + 19 n q - 15 n s + 12 n q - 14 n q s + 2 n s + 2 q - 2 q s 2 3 2 2 2 - q s + s - 2 n + 3 n q - 7 n s + 2 q - 4 q s + s - 4 n - q - s - 1) /d \ / 2 2 |-- A[n](r, s)| / ((n - s + 1) (n - 2 n s + s + 2 n - 2 s + 1) \ds / / 4 3 3 2 2 2 2 3 3 (n + q - s)) - (n + n q + n s + 2 n q s - n s + n q s - n s + 4 n 2 2 2 2 3 2 + 3 n q + 3 n s + 4 n q s - 2 n s + q s - s + 6 n + 3 n q + 3 n s / 2 \ 2 |d | / + 2 q s - s + 4 n + q + s + 1) |--- A[n](r, s)| / ( | 2 | / \ds / 3 2 2 3 2 2 n - 3 n s + 3 n s - s + 3 n - 6 n s + 3 s + 3 n - 3 s + 1) + 2 2 2 3 / (n + 2 n q - 2 n s + q - 2 q s + s - n - q + s) q A[n + 1](r, s) / ( / 2 2 2 (n + 2 n q - 2 n s + q - 2 q s + s + 2 n + 2 q - 2 s + 1) 3 2 2 3 2 2 (n - 3 n s + 3 n s - s + 3 n - 6 n s + 3 s + 3 n - 3 s + 1)) + ( 3 2 2 3 2 2 4 3 2 2 5 3 n q + 9 n q - 9 n q s + 9 n q - 18 n q s + 9 n q s + 3 q 4 3 2 2 3 3 2 2 2 3 - 9 q s + 9 q s - 3 q s - 3 n q - 9 n q + 9 n q s - 9 n q 2 2 4 3 2 2 3 3 2 + 18 n q s - 9 n q s - 3 q + 9 q s - 9 q s + 3 q s + n + n q 2 2 2 3 2 2 3 2 - 3 n s - 3 n q - 2 n q s + 3 n s - 3 q + 3 q s + q s - s + n 2 2 /d \ + 5 n q - 2 n s + 3 q - 5 q s + s - n + q + s - 1) |-- A[n + 1](r, s)| \ds / / 2 2 / ((n + q - s) (n - 2 n s + s + 2 n - 2 s + 1) / 2 2 2 2 (n + 2 n q - 2 n s + q - 2 q s + s + 2 n + 2 q - 2 s + 1)) + (3 n q 2 3 2 2 2 2 + 6 n q - 6 n q s + 3 q - 6 q s + 3 q s - 3 n - 6 n q + 6 n s - 3 q / 2 \ 2 |d | + 6 q s - 3 s - 2 n - 3 q + 2 s + 1) |--- A[n + 1](r, s)|/((n - s + 1) | 2 | \ds / / 3 \ |d | (n + q - s) (n + q - s + 1)) + |--- A[n + 1](r, s)| = 0 | 3 | \ds / and in Maple notation -(n^3*p+n^3*s-2*n^2*p^2-2*n^2*p*s+n*p^3+n*p^2*s-n^3+7*n^2*p+5*n^2*s-7*n*p^2-6*n *p*s+p^3+p^2*s-5*n^2+14*n*p+8*n*s-5*p^2-4*p*s-8*n+8*p+4*s-4)/(n^3+3*n^2*p+3*n*p ^2+p^3)*A[n](r,s)+(2*n^3*p^3-3*n^3*p^2*r+3*n^3*p^2*s-6*n^3*p*r*s+n^3*r^3+3*n^3* r^2*s-2*n^2*p^4-4*n^2*p^3*s+6*n^2*p^2*r^2+6*n^2*p^2*r*s-4*n^2*p*r^3-2*n^2*r^3*s +3*n*p^4*r+n*p^4*s-6*n*p^3*r^2+3*n*p^2*r^3-3*n*p^2*r^2*s+2*n*p*r^3*s-9*n^3*p^2+ 9*n^3*p*r-9*n^3*p*s+9*n^3*r*s+18*n^2*p^3-9*n^2*p^2*r+27*n^2*p^2*s-18*n^2*p*r^2-\ 36*n^2*p*r*s+9*n^2*r^3+9*n^2*r^2*s-4*n*p^4-18*n*p^3*r-14*n*p^3*s+39*n*p^2*r^2+ 12*n*p^2*r*s-17*n*p*r^3+9*n*p*r^2*s-7*n*r^3*s+3*p^4*r+p^4*s-6*p^3*r^2+3*p^2*r^3 -3*p^2*r^2*s+2*p*r^3*s+13*n^3*p-6*n^3*r+7*n^3*s-53*n^2*p^2+27*n^2*p*r-53*n^2*p* s+12*n^2*r^2+39*n^2*r*s+30*n*p^3+30*n*p^2*r+58*n*p^2*s-75*n*p*r^2-54*n*p*r*s+22 *n*r^3+3*n*r^2*s-2*p^4-18*p^3*r-10*p^3*s+33*p^2*r^2+6*p^2*r*s-13*p*r^3+9*p*r^2* s-5*r^3*s-6*n^3+63*n^2*p-18*n^2*r+33*n^2*s-79*n*p^2-9*n*p*r-91*n*p*s+42*n*r^2+ 51*n*r*s+14*p^3+36*p^2*r+34*p^2*s-57*p*r^2-24*p*r*s+14*r^3-3*r^2*s-26*n^2+87*n* p-6*n*r+49*n*s-35*p^2-27*p*r-47*p*s+30*r^2+21*r*s-34*n+37*p+6*r+23*s-14)/(n^3*p -n^3*r+3*n^2*p^2-3*n^2*p*r+3*n*p^3-3*n*p^2*r+p^4-p^3*r-2*n^3-6*n^2*p-6*n*p^2-2* p^3)/(p-r-2)*diff(A[n](r,s),r)-(n^3*p^4-n^3*p^3*r+3*n^3*p^3*s-3*n^3*p^2*r^2-9*n ^3*p^2*r*s+5*n^3*p*r^3+9*n^3*p*r^2*s-2*n^3*r^4-3*n^3*r^3*s-4*n^2*p^4*r-2*n^2*p^ 4*s+10*n^2*p^3*r^2+2*n^2*p^3*r*s-6*n^2*p^2*r^3+6*n^2*p^2*r^2*s-2*n^2*p*r^4-10*n ^2*p*r^3*s+2*n^2*r^5+4*n^2*r^4*s+3*n*p^4*r^2+2*n*p^4*r*s-9*n*p^3*r^3-5*n*p^3*r^ 2*s+9*n*p^2*r^4+3*n*p^2*r^3*s-3*n*p*r^5+n*p*r^4*s-n*r^5*s-7*n^3*p^3+6*n^3*p^2*r -15*n^3*p^2*s+9*n^3*p*r^2+30*n^3*p*r*s-8*n^3*r^3-15*n^3*r^2*s+3*n^2*p^4+25*n^2* p^3*r+23*n^2*p^3*s-63*n^2*p^2*r^2-39*n^2*p^2*r*s+39*n^2*p*r^3+9*n^2*p*r^2*s-4*n ^2*r^4+7*n^2*r^3*s-8*n*p^4*r-4*n*p^4*s-n*p^3*r^2-10*n*p^3*r*s+36*n*p^2*r^3+39*n *p^2*r^2*s-37*n*p*r^4-32*n*p*r^3*s+10*n*r^5+7*n*r^4*s+3*p^4*r^2+2*p^4*r*s-9*p^3 *r^3-5*p^3*r^2*s+9*p^2*r^4+3*p^2*r^3*s-3*p*r^5+p*r^4*s-r^5*s+17*n^3*p^2-11*n^3* p*r+23*n^3*p*s-6*n^3*r^2-23*n^3*r*s-21*n^2*p^3-50*n^2*p^2*r-79*n^2*p^2*s+117*n^ 2*p*r^2+112*n^2*p*r*s-46*n^2*r^3-33*n^2*r^2*s+3*n*p^4+53*n*p^3*r+37*n*p^3*s-66* n*p^2*r^2-17*n*p^2*r*s-16*n*p*r^3-54*n*p*r^2*s+26*n*r^4+34*n*r^3*s-4*p^4*r-2*p^ 4*s-11*p^3*r^2-12*p^3*r*s+42*p^2*r^3+33*p^2*r^2*s-35*p*r^4-22*p*r^3*s+8*r^5+3*r ^4*s-17*n^3*p+6*n^3*r-11*n^3*s+51*n^2*p^2+35*n^2*p*r+103*n^2*p*s-64*n^2*r^2-81* n^2*r*s-21*n*p^3-118*n*p^2*r-113*n*p^2*s+156*n*p*r^2+100*n*p*r*s-28*n*r^3+2*n*r ^2*s+p^4+27*p^3*r+17*p^3*s-6*p^2*r^2+13*p^2*r*s-50*p*r^3-54*p*r^2*s+28*r^4+24*r ^3*s+6*n^3-51*n^2*p-6*n^2*r-45*n^2*s+51*n*p^2+103*n*p*r+137*n*p*s-92*n*r^2-81*n *r*s-7*p^3-62*p^2*r-49*p^2*s+48*p*r^2+18*p*r*s+10*r^3+20*r^2*s+18*n^2-51*n*p-30 *n*r-57*n*s+17*p^2+57*p*r+57*p*s-34*r^2-23*r*s+18*n-17*p-18*r-23*s+6)/(p^3-3*p^ 2*r+3*p*r^2-r^3-7*p^2+14*p*r-7*r^2+16*p-16*r-12)/(n^3+3*n^2*p+3*n*p^2+p^3)*diff (diff(A[n](r,s),r),r)+(n^3*p^2*r+n^3*p^2*s-2*n^3*p*r^2-2*n^3*p*r*s+n^3*r^3+n^3* r^2*s-2*n^2*p^2*r^2-2*n^2*p^2*r*s+4*n^2*p*r^3+4*n^2*p*r^2*s-2*n^2*r^4-2*n^2*r^3 *s+n*p^2*r^3+n*p^2*r^2*s-2*n*p*r^4-2*n*p*r^3*s+n*r^5+n*r^4*s-2*n^3*p*r-2*n^3*p* s+2*n^3*r^2+2*n^3*r*s+3*n^2*p^2*r+3*n^2*p^2*s-2*n^2*p*r^2-2*n^2*p*r*s-n^2*r^3-n ^2*r^2*s-4*n*p^2*r^2-4*n*p^2*r*s+6*n*p*r^3+6*n*p*r^2*s-2*n*r^4-2*n*r^3*s+p^2*r^ 3+p^2*r^2*s-2*p*r^4-2*p*r^3*s+r^5+r^4*s+n^3*r+n^3*s-6*n^2*p*r-6*n^2*p*s+4*n^2*r ^2+4*n^2*r*s+3*n*p^2*r+3*n*p^2*s+2*n*p*r^2+2*n*p*r*s-4*n*r^3-4*n*r^2*s-2*p^2*r^ 2-2*p^2*r*s+2*p*r^3+2*p*r^2*s+3*n^2*r+3*n^2*s-6*n*p*r-6*n*p*s+2*n*r^2+2*n*r*s+p ^2*r+p^2*s+2*p*r^2+2*p*r*s-2*r^3-2*r^2*s+3*n*r+3*n*s-2*p*r-2*p*s+r+s)/(n^3*p-n^ 3*r+3*n^2*p^2-3*n^2*p*r+3*n*p^3-3*n*p^2*r+p^4-p^3*r-2*n^3-6*n^2*p-6*n*p^2-2*p^3 )/(-3+p-r)*diff(diff(diff(A[n](r,s),r),r),r)+A[n+1](r,s)-(3*n^3*p^2-6*n^3*p*r+3 *n^3*r^2+6*n^2*p^3-9*n^2*p^2*r+3*n^2*r^3+3*n*p^4-9*n*p^2*r^2+6*n*p*r^3+3*p^4*r-\ 6*p^3*r^2+3*p^2*r^3-9*n^3*p+9*n^3*r-18*n^2*p^2+9*n^2*p*r+9*n^2*r^2-6*n*p^3-18*n *p^2*r+27*n*p*r^2-3*n*r^3+3*p^4-18*p^3*r+18*p^2*r^2-3*p*r^3+7*n^3+12*n^2*p+9*n^ 2*r-6*n*p^2+36*n*p*r-9*n*r^2-12*p^3+30*p^2*r-12*p*r^2+r^3+3*n^2+15*n*p-9*n*r+15 *p^2-15*p*r+3*r^2-3*n-6*p+3*r+1)/(n^3*p^2-2*n^3*p*r+n^3*r^2+3*n^2*p^3-6*n^2*p^2 *r+3*n^2*p*r^2+3*n*p^4-6*n*p^3*r+3*n*p^2*r^2+p^5-2*p^4*r+p^3*r^2-4*n^3*p+4*n^3* r-12*n^2*p^2+12*n^2*p*r-12*n*p^3+12*n*p^2*r-4*p^4+4*p^3*r+4*n^3+12*n^2*p+12*n*p ^2+4*p^3)*diff(A[n+1](r,s),r)+(3*n^3*p^3-9*n^3*p^2*r+9*n^3*p*r^2-3*n^3*r^3+3*n^ 2*p^4-3*n^2*p^3*r-9*n^2*p^2*r^2+15*n^2*p*r^3-6*n^2*r^4+6*n*p^4*r-15*n*p^3*r^2+9 *n*p^2*r^3+3*n*p*r^4-3*n*r^5+3*p^4*r^2-9*p^3*r^3+9*p^2*r^4-3*p*r^5-15*n^3*p^2+ 30*n^3*p*r-15*n^3*r^2-12*n^2*p^3-9*n^2*p^2*r+54*n^2*p*r^2-33*n^2*r^3+6*n*p^4-48 *n*p^3*r+63*n*p^2*r^2-6*n*p*r^3-15*n*r^4+6*p^4*r-36*p^3*r^2+57*p^2*r^3-30*p*r^4 +3*r^5+23*n^3*p-23*n^3*r+6*n^2*p^2+57*n^2*p*r-63*n^2*r^2-33*n*p^3+111*n*p^2*r-\ 54*n*p*r^2-24*n*r^3+3*p^4-45*p^3*r+123*p^2*r^2-100*p*r^3+19*r^4-11*n^3+18*n^2*p -51*n^2*r+57*n*p^2-78*n*p*r-12*n*r^2-18*p^3+111*p^2*r-150*p*r^2+46*r^3-15*n^2-\ 33*n*p+3*n*r+36*p^2-105*p*r+54*r^2+3*n-28*p+31*r+7)/(-3+p-r)/(n^3*p^2-2*n^3*p*r +n^3*r^2+3*n^2*p^3-6*n^2*p^2*r+3*n^2*p*r^2+3*n*p^4-6*n*p^3*r+3*n*p^2*r^2+p^5-2* p^4*r+p^3*r^2-4*n^3*p+4*n^3*r-12*n^2*p^2+12*n^2*p*r-12*n*p^3+12*n*p^2*r-4*p^4+4 *p^3*r+4*n^3+12*n^2*p+12*n*p^2+4*p^3)*diff(diff(A[n+1](r,s),r),r)-(n^3*p^2-2*n^ 3*p*r+n^3*r^2+3*n^2*p^2*r-6*n^2*p*r^2+3*n^2*r^3+3*n*p^2*r^2-6*n*p*r^3+3*n*r^4+p ^2*r^3-2*p*r^4+r^5-2*n^3*p+2*n^3*r+3*n^2*p^2-12*n^2*p*r+9*n^2*r^2+6*n*p^2*r-18* n*p*r^2+12*n*r^3+3*p^2*r^2-8*p*r^3+5*r^4+n^3-6*n^2*p+9*n^2*r+3*n*p^2-18*n*p*r+ 18*n*r^2+3*p^2*r-12*p*r^2+10*r^3+3*n^2-6*n*p+12*n*r+p^2-8*p*r+10*r^2+3*n-2*p+5* r+1)/(n^3*p-n^3*r+3*n^2*p^2-3*n^2*p*r+3*n*p^3-3*n*p^2*r+p^4-p^3*r-2*n^3-6*n^2*p -6*n*p^2-2*p^3)/(-3+p-r)*diff(diff(diff(A[n+1](r,s),r),r),r) = 0 -(4*n^5+12*n^4*q-8*n^4*s+13*n^3*q^2-16*n^3*q*s+4*n^3*s^2+6*n^2*q^3-10*n^2*q^2*s +4*n^2*q*s^2+n*q^4-2*n*q^3*s+n*q^2*s^2+8*n^4+22*n^3*q-16*n^3*s+21*n^2*q^2-28*n^ 2*q*s+8*n^2*s^2+8*n*q^3-14*n*q^2*s+6*n*q*s^2+q^4-2*q^3*s+q^2*s^2+n^3+8*n^2*q-10 *n^2*s+8*n*q^2-14*n*q*s+5*n*s^2+2*q^3-4*q^2*s+2*q*s^2-7*n^2-4*n*q-2*n*s-2*q*s+s ^2-5*n-2*q-1)/(n-s+1)/(n^2-2*n*s+s^2+2*n-2*s+1)/(n+q-s)*A[n](r,s)+(4*n^5+10*n^4 *q-4*n^4*s+8*n^3*q^2-2*n^3*q*s-4*n^3*s^2+2*n^2*q^3+4*n^2*q^2*s-10*n^2*q*s^2+4*n ^2*s^3+2*n*q^3*s-4*n*q^2*s^2+2*n*q*s^3+11*n^4+25*n^3*q-13*n^3*s+18*n^2*q^2-12*n ^2*q*s-3*n^2*s^2+4*n*q^3+2*n*q^2*s-11*n*q*s^2+5*n*s^3+2*q^3*s-4*q^2*s^2+2*q*s^3 +8*n^3+19*n^2*q-15*n^2*s+12*n*q^2-14*n*q*s+2*n*s^2+2*q^3-2*q^2*s-q*s^2+s^3-2*n^ 2+3*n*q-7*n*s+2*q^2-4*q*s+s^2-4*n-q-s-1)/(n-s+1)/(n^2-2*n*s+s^2+2*n-2*s+1)/(n+q -s)*diff(A[n](r,s),s)-(n^4+n^3*q+n^3*s+2*n^2*q*s-n^2*s^2+n*q*s^2-n*s^3+4*n^3+3* n^2*q+3*n^2*s+4*n*q*s-2*n*s^2+q*s^2-s^3+6*n^2+3*n*q+3*n*s+2*q*s-s^2+4*n+q+s+1)/ (n^3-3*n^2*s+3*n*s^2-s^3+3*n^2-6*n*s+3*s^2+3*n-3*s+1)*diff(diff(A[n](r,s),s),s) +(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2-n-q+s)*q^3/(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2+2*n+2 *q-2*s+1)/(n^3-3*n^2*s+3*n*s^2-s^3+3*n^2-6*n*s+3*s^2+3*n-3*s+1)*A[n+1](r,s)+(3* n^3*q^2+9*n^2*q^3-9*n^2*q^2*s+9*n*q^4-18*n*q^3*s+9*n*q^2*s^2+3*q^5-9*q^4*s+9*q^ 3*s^2-3*q^2*s^3-3*n^3*q-9*n^2*q^2+9*n^2*q*s-9*n*q^3+18*n*q^2*s-9*n*q*s^2-3*q^4+ 9*q^3*s-9*q^2*s^2+3*q*s^3+n^3+n^2*q-3*n^2*s-3*n*q^2-2*n*q*s+3*n*s^2-3*q^3+3*q^2 *s+q*s^2-s^3+n^2+5*n*q-2*n*s+3*q^2-5*q*s+s^2-n+q+s-1)/(n+q-s)/(n^2-2*n*s+s^2+2* n-2*s+1)/(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2+2*n+2*q-2*s+1)*diff(A[n+1](r,s),s)+(3*n ^2*q+6*n*q^2-6*n*q*s+3*q^3-6*q^2*s+3*q*s^2-3*n^2-6*n*q+6*n*s-3*q^2+6*q*s-3*s^2-\ 2*n-3*q+2*s+1)/(n-s+1)/(n+q-s)/(n+q-s+1)*diff(diff(A[n+1](r,s),s),s)+diff(diff( diff(A[n+1](r,s),s),s),s) = 0 ------------------------------------------------- This took, 0.365, seconds. -------------------------------------------- Theorem: define the Abel-sum type sequence by n ----- \ 3 3 (k - 1 + p) A[n](r, s) = ) binomial(n, k) binomial(n + k, k) (r + k) / ----- k = 0 (n - k + q) k (s - k) x and in Maple notation A[n](r,s) = Sum(binomial(n,k)^3*binomial(n+k,k)^3*(r+k)^(k-1+p)*(s-k)^(n-k+q)*x ^k,k = 0 .. n) Then we have the following two differential-recurrence equations, relating \ A[n](r,s) and A[n+1](r,s) , the first one with respect to r, the second \ with respect to s 3 4 3 3 3 3 3 2 2 3 2 3 3 3 2 (n p - 3 n p r + n p s + 3 n p r - 3 n p r s - n p r + 3 n p r s 3 3 2 5 2 4 2 4 2 3 2 2 3 - n r s - 3 n p + 9 n p r - 3 n p s - 9 n p r + 9 n p r s 2 2 3 2 2 2 2 3 6 5 5 + 3 n p r - 9 n p r s + 3 n p r s + 3 n p - 9 n p r + 3 n p s 4 2 4 3 3 3 2 2 3 7 + 9 n p r - 9 n p r s - 3 n p r + 9 n p r s - 3 n p r s - p 6 6 5 2 5 4 3 4 2 3 3 + 3 p r - p s - 3 p r + 3 p r s + p r - 3 p r s + p r s 3 3 3 2 3 2 3 2 3 3 3 - 7 n p + 15 n p r - 6 n p s - 9 n p r + 12 n p r s + n r 3 2 2 4 2 3 2 3 2 2 2 - 6 n r s + 27 n p - 63 n p r + 24 n p s + 45 n p r 2 2 2 3 2 2 2 3 5 - 54 n p r s - 9 n p r + 36 n p r s - 6 n r s - 33 n p 4 4 3 2 3 2 3 + 81 n p r - 30 n p s - 63 n p r + 72 n p r s + 15 n p r 2 2 3 6 5 5 4 2 - 54 n p r s + 12 n p r s + 13 p - 33 p r + 12 p s + 27 p r 4 3 3 3 2 2 3 3 2 3 - 30 p r s - 7 p r + 24 p r s - 6 p r s + 18 n p - 24 n p r 3 3 2 3 2 3 2 2 2 2 + 12 n p s + 6 n r - 12 n r s - 96 n p + 162 n p r - 72 n p s 2 2 2 2 3 2 2 4 3 - 72 n p r + 108 n p r s + 6 n r - 36 n r s + 150 n p - 288 n p r 3 2 2 2 3 2 + 120 n p s + 162 n p r - 216 n p r s - 24 n p r + 108 n p r s 3 5 4 4 3 2 3 - 12 n r s - 72 p + 150 p r - 60 p s - 96 p r + 120 p r s 2 3 2 2 3 3 3 3 + 18 p r - 72 p r s + 12 p r s - 20 n p + 12 n r - 8 n s 2 2 2 2 2 2 2 3 + 168 n p - 180 n p r + 96 n p s + 36 n r - 72 n r s - 360 n p 2 2 2 3 2 + 504 n p r - 240 n p s - 180 n p r + 288 n p r s + 12 n r - 72 n r s 4 3 3 2 2 2 3 + 220 p - 360 p r + 160 p s + 168 p r - 240 p r s - 20 p r 2 3 3 2 2 2 2 + 96 p r s - 8 r s + 8 n - 144 n p + 72 n r - 48 n s + 480 n p 2 3 2 - 432 n p r + 240 n p s + 72 n r - 144 n r s - 400 p + 480 p r 2 2 3 2 2 - 240 p s - 144 p r + 240 p r s + 8 r - 48 r s + 48 n - 336 n p 2 2 + 144 n r - 96 n s + 432 p - 336 p r + 192 p s + 48 r - 96 r s + 96 n 3 4 3 3 - 256 p + 96 r - 64 s + 64) A[n](r, s)/(%2) - (3 n p - 8 n p r 3 3 3 2 2 3 2 3 2 3 4 3 3 + 4 n p s + 6 n p r - 12 n p r s + 12 n p r s - n r - 4 n r s 2 5 2 4 2 4 2 3 2 2 3 - 6 n p + 12 n p r - 9 n p s + 24 n p r s - 12 n p r 2 2 2 2 4 2 4 6 5 4 2 - 18 n p r s + 6 n p r + 3 n r s + 3 n p + 6 n p s - 18 n p r 4 3 3 2 4 2 3 4 - 12 n p r s + 24 n p r - 9 n p r + 12 n p r s - 6 n p r s 6 6 5 2 4 3 4 2 3 4 3 3 - 4 p r - p s + 12 p r - 12 p r + 6 p r s + 4 p r - 8 p r s 2 4 3 3 3 2 3 2 3 2 + 3 p r s - 18 n p + 36 n p r - 18 n p s - 18 n p r 3 3 2 2 4 2 3 2 3 + 36 n p r s - 18 n r s + 54 n p - 96 n p r + 66 n p s 2 2 2 2 2 2 3 2 2 2 4 + 18 n p r - 144 n p r s + 36 n p r + 90 n p r s - 12 n r 2 3 5 4 4 3 2 - 12 n r s - 39 n p + 24 n p r - 63 n p s + 108 n p r 3 2 3 2 2 4 3 + 120 n p r s - 132 n p r - 36 n p r s + 39 n p r - 36 n p r s 4 6 5 5 4 2 4 3 3 + 15 n r s + 3 p + 36 p r + 15 p s - 108 p r - 12 p r s + 96 p r 3 2 2 4 2 3 4 3 2 3 - 36 p r s - 27 p r + 48 p r s - 15 p r s + 40 n p - 52 n p r 3 3 2 3 2 3 2 2 2 2 + 28 n p s + 12 n r - 28 n r s - 186 n p + 264 n p r - 174 n p s 2 2 2 2 3 2 2 4 - 54 n p r + 264 n p r s - 24 n r - 90 n r s + 198 n p 3 3 2 2 2 3 - 168 n p r + 252 n p s - 216 n p r - 408 n p r s + 228 n p r 2 4 3 5 4 4 + 144 n p r s - 42 n r + 12 n r s - 33 p - 120 p r - 87 p s 3 2 3 2 3 2 2 4 3 + 372 p r + 96 p r s - 280 p r + 60 p r s + 61 p r - 88 p r s 4 3 3 3 2 2 2 + 19 r s - 39 n p + 24 n r - 15 n s + 309 n p - 300 n p r 2 2 2 2 3 2 2 + 201 n p s + 36 n r - 156 n r s - 507 n p + 420 n p r - 483 n p s 2 3 2 4 3 + 162 n p r + 564 n p r s - 120 n r - 126 n r s + 147 p + 172 p r 3 2 2 2 3 2 4 + 253 p s - 606 p r - 276 p r s + 348 p r - 6 p r s - 46 r 3 3 2 2 2 2 + 44 r s + 14 n - 249 n p + 120 n r - 87 n s + 696 n p - 444 n p r 2 3 2 2 2 + 450 n p s - 36 n r - 276 n r s - 339 p - 72 p r - 393 p s + 462 p r 3 2 2 + 336 p r s - 152 r - 30 r s + 78 n - 489 n p + 168 n r - 165 n s 2 2 + 427 p - 52 p r + 313 p s - 132 r - 148 r s + 138 n - 279 p + 40 r /d \ 3 6 3 5 3 5 - 101 s + 74) |-- A[n](r, s)|/(%2) + (3 n p - 12 n p r + 6 n p s \dr / 3 4 2 3 4 3 3 2 3 2 4 3 2 3 + 15 n p r - 30 n p r s + 60 n p r s - 15 n p r - 60 n p r s 3 5 3 4 3 6 3 5 2 7 2 6 + 12 n p r + 30 n p r s - 3 n r - 6 n r s - 3 n p + 3 n p r 2 6 2 5 2 2 5 2 4 3 2 4 2 - 9 n p s + 27 n p r + 36 n p r s - 75 n p r - 45 n p r s 2 3 4 2 2 5 2 2 4 2 6 2 5 + 75 n p r - 27 n p r + 45 n p r s - 3 n p r - 36 n p r s 2 7 2 6 7 7 6 2 6 + 3 n r + 9 n r s + 9 n p r + 3 n p s - 36 n p r - 3 n p r s 5 3 5 2 4 3 3 5 3 4 + 45 n p r - 27 n p r s + 75 n p r s - 45 n p r - 75 n p r s 2 6 2 5 7 6 7 7 2 + 36 n p r + 27 n p r s - 9 n p r + 3 n p r s - 3 n r s - 6 p r 7 6 3 6 2 5 4 5 3 4 5 - 3 p r s + 30 p r + 12 p r s - 60 p r - 15 p r s + 60 p r 3 6 3 5 2 7 2 6 7 3 5 - 30 p r + 15 p r s + 6 p r - 12 p r s + 3 p r s - 33 n p 3 4 3 4 3 3 2 3 3 3 2 3 + 111 n p r - 54 n p s - 114 n p r + 216 n p r s + 6 n p r 3 2 2 3 4 3 3 3 5 3 4 - 324 n p r s + 51 n p r + 216 n p r s - 21 n r - 54 n r s 2 6 2 5 2 5 2 4 2 2 4 + 48 n p - 72 n p r + 117 n p s - 198 n p r - 423 n p r s 2 3 3 2 3 2 2 2 4 2 2 3 + 552 n p r + 522 n p r s - 468 n p r - 198 n p r s 2 5 2 4 2 6 2 5 7 6 + 144 n p r - 63 n p r s - 6 n r + 45 n r s - 6 n p - 111 n p r 6 5 2 5 4 3 4 2 - 57 n p s + 459 n p r + 108 n p r s - 582 n p r + 153 n p r s 3 4 3 3 2 5 2 4 6 + 168 n p r - 552 n p r s + 189 n p r + 513 n p r s - 141 n p r 5 7 6 7 7 6 2 - 180 n p r s + 24 n r + 15 n r s + 9 p r + 3 p s + 42 p r 6 5 3 5 2 4 4 4 3 + 36 p r s - 291 p r - 162 p r s + 564 p r + 219 p r s 3 5 3 4 2 6 2 5 7 6 - 501 p r - 81 p r s + 210 p r - 54 p r s - 33 p r + 48 p r s 7 3 4 3 3 3 3 3 2 2 - 9 r s + 146 n p - 395 n p r + 189 n p s + 309 n p r 3 2 3 3 3 2 3 4 3 3 - 567 n p r s - 17 n p r + 567 n p r s - 43 n r - 189 n r s 2 5 2 4 2 4 2 3 2 2 3 - 309 n p + 507 n p r - 600 n p s + 495 n p r + 1833 n p r s 2 2 3 2 2 2 2 4 2 3 - 1437 n p r - 1899 n p r s + 906 n p r + 699 n p r s 2 5 2 4 6 5 5 - 162 n r - 33 n r s + 87 n p + 522 n p r + 426 n p s 4 2 4 3 3 3 2 2 4 - 2277 n p r - 930 n p r s + 2715 n p r + 27 n p r s - 1008 n p r 2 3 5 4 6 5 + 1239 n p r s - 153 n p r - 969 n p r s + 114 n r + 207 n r s 7 6 6 5 2 5 4 3 - 3 p - 114 p r - 48 p s + 12 p r - 138 p r s + 1009 p r 4 2 3 4 3 3 2 5 2 4 + 810 p r s - 1960 p r - 1089 p r s + 1479 p r + 507 p r s 6 5 7 6 3 3 3 2 - 469 p r - 9 p r s + 46 r - 33 r s - 332 n p + 675 n p r 3 2 3 2 3 3 3 3 2 - 321 n p s - 354 n p r + 642 n p r s + 11 n r - 321 n r s 2 4 2 3 2 3 2 2 2 2 2 + 1044 n p - 1617 n p r + 1563 n p s - 450 n p r - 3726 n p r s 2 3 2 2 2 4 2 3 5 + 1575 n p r + 2763 n p r s - 552 n r - 600 n r s - 519 n p 4 4 3 2 3 2 3 - 1137 n p r - 1644 n p s + 5616 n p r + 3450 n p r s - 5799 n p r 2 2 4 3 5 4 - 1449 n p r s + 1893 n p r - 876 n p r s - 54 n r + 519 n r s 6 5 5 4 2 4 3 3 + 42 p + 582 p r + 315 p s - 852 p r + 69 p r s - 1357 p r 3 2 2 4 2 3 5 4 - 1863 p r s + 3054 p r + 2346 p r s - 1791 p r - 954 p r s 6 5 3 2 3 3 3 2 + 322 r + 87 r s + 409 n p - 553 n p r + 265 n p s + 144 n r 3 2 3 2 2 2 2 2 2 - 265 n r s - 2007 n p + 2604 n p r - 2190 n p s + 18 n p r 2 2 3 2 2 4 3 + 3585 n p r s - 615 n r - 1395 n r s + 1650 n p + 984 n p r 3 2 2 2 3 2 + 3570 n p s - 7245 n p r - 6330 n p r s + 5733 n p r + 2745 n p r s 4 3 5 4 4 3 2 - 1122 n r + 15 n r s - 243 p - 1533 p r - 1098 p s + 2985 p r 3 2 3 2 2 4 3 5 + 822 p r s + 74 p r + 1932 p r s - 2021 p r - 2203 p r s + 738 r 4 3 3 3 2 2 2 + 547 r s - 259 n p + 174 n r - 85 n s + 2202 n p - 2055 n p r 2 2 2 2 3 2 + 1572 n p s + 108 n r - 1317 n r s - 3018 n p + 258 n p r 2 2 3 2 - 4392 n p s + 4617 n p r + 5640 n p r s - 2112 n r - 1503 n r s 4 3 3 2 2 2 3 + 752 p + 2206 p r + 2196 p s - 4536 p r - 2196 p r s + 1277 p r 2 4 3 3 2 2 2 - 624 p r s + 386 r + 709 r s + 66 n - 1281 n p + 630 n r - 453 n s 2 2 3 + 3177 n p - 939 n p r + 2853 n p s - 1134 n r - 1947 n r s - 1343 p 2 2 2 3 2 - 1671 p r - 2523 p s + 3237 p r + 2193 p r s - 742 r - 123 r s 2 2 + 306 n - 1785 n p + 414 n r - 759 n s + 1384 p + 563 p r + 1546 p s / 2 \ 2 |d | - 882 r - 787 r s + 414 n - 763 p - 42 r - 391 s + 174) |--- A[n](r, s)|/ | 2 | \dr / 3 7 3 6 3 6 3 5 2 (%1 (-3 + p - r)) - (n p - 3 n p r + 4 n p s - 3 n p r 3 5 3 4 3 3 4 2 3 3 4 3 3 3 - 24 n p r s + 25 n p r + 60 n p r s - 45 n p r - 80 n p r s 3 2 5 3 2 4 3 6 3 5 3 7 + 39 n p r + 60 n p r s - 17 n p r - 24 n p r s + 3 n r 3 6 2 7 2 7 2 6 2 2 6 + 4 n r s - 6 n p r - 3 n p s + 30 n p r + 9 n p r s 2 5 3 2 5 2 2 4 4 2 4 3 2 3 5 - 54 n p r + 9 n p r s + 30 n p r - 75 n p r s + 30 n p r 2 3 4 2 2 6 2 2 5 2 7 + 135 n p r s - 54 n p r - 117 n p r s + 30 n p r 2 6 2 8 2 7 7 2 7 6 3 + 51 n p r s - 6 n r - 9 n r s + 9 n p r + 6 n p r s - 51 n p r 6 2 5 4 5 3 4 5 4 4 - 30 n p r s + 117 n p r + 54 n p r s - 135 n p r - 30 n p r s 3 6 3 5 2 7 2 6 8 + 75 n p r - 30 n p r s - 9 n p r + 54 n p r s - 9 n p r 7 9 8 7 3 7 2 6 4 - 30 n p r s + 3 n r + 6 n r s - 4 p r - 3 p r s + 24 p r 6 3 5 5 5 4 4 6 4 5 3 7 + 17 p r s - 60 p r - 39 p r s + 80 p r + 45 p r s - 60 p r 3 6 2 8 2 7 9 8 9 3 6 - 25 p r s + 24 p r + 3 p r s - 4 p r + 3 p r s - r s - 14 n p 3 5 3 5 3 4 2 3 4 3 3 3 + 38 n p r - 46 n p s + 20 n p r + 230 n p r s - 180 n p r 3 3 2 3 2 4 3 2 3 3 5 - 460 n p r s + 250 n p r + 460 n p r s - 146 n p r 3 4 3 6 3 5 2 7 2 6 - 230 n p r s + 32 n r + 46 n r s + 3 n p + 75 n p r 2 6 2 5 2 2 5 2 4 3 + 54 n p s - 375 n p r - 186 n p r s + 645 n p r 2 4 2 2 3 4 2 3 3 2 2 5 + 120 n p r s - 435 n p r + 300 n p r s - 3 n p r 2 2 4 2 6 2 5 2 7 2 6 - 570 n p r s + 135 n p r + 366 n p r s - 45 n r - 84 n r s 7 7 6 2 6 5 3 - 12 n p r - 6 n p s - 66 n p r - 66 n p r s + 510 n p r 5 2 4 4 4 3 3 5 + 384 n p r s - 1140 n p r - 720 n p r s + 1200 n p r 3 4 2 6 2 5 7 6 + 570 n p r s - 618 n p r - 114 n p r s + 126 n p r - 84 n p r s 7 7 2 7 6 3 6 2 5 4 + 36 n r s + 9 p r + 6 p r s + 5 p r + 12 p r s - 173 p r 5 3 4 5 4 4 3 6 3 5 - 152 p r s + 475 p r + 370 p r s - 585 p r - 410 p r s 2 7 2 6 8 7 9 8 + 371 p r + 224 p r s - 115 p r - 52 p r s + 13 r + 2 r s 3 5 3 4 3 4 3 3 2 3 3 + 80 n p - 190 n p r + 210 n p s - 40 n p r - 840 n p r s 3 2 3 3 2 2 3 4 3 3 3 5 + 460 n p r + 1260 n p r s - 440 n p r - 840 n p r s + 130 n r 3 4 2 6 2 5 2 5 2 4 2 + 210 n r s - 42 n p - 366 n p r - 378 n p s + 1830 n p r 2 4 2 3 3 2 3 2 2 2 4 + 1260 n p r s - 2820 n p r - 1260 n p r s + 1770 n p r 2 5 2 4 2 6 2 5 7 - 318 n p r + 630 n p r s - 54 n r - 252 n r s + 3 n p 6 6 5 2 5 4 3 + 159 n p r + 96 n p s - 21 n p r + 180 n p r s - 1755 n p r 4 2 3 4 3 3 2 5 - 1710 n p r s + 3945 n p r + 3120 n p r s - 3543 n p r 2 4 6 5 7 6 - 2340 n p r s + 1401 n p r + 684 n p r s - 189 n r - 30 n r s 7 7 6 2 6 5 3 5 2 - 6 p r - 3 p s - 96 p r - 75 p r s + 244 p r + 135 p r s 4 4 4 3 3 5 3 4 2 6 + 220 p r + 345 p r s - 1190 p r - 1125 p r s + 1376 p r 2 5 7 6 8 7 3 4 + 1143 p r s - 664 p r - 495 p r s + 116 r + 75 r s - 242 n p 3 3 3 3 3 2 2 3 2 3 3 + 480 n p r - 488 n p s + 12 n p r + 1464 n p r s - 496 n p r 3 2 3 4 3 3 2 5 2 4 - 1464 n p r s + 246 n r + 488 n r s + 240 n p + 882 n p r 2 4 2 3 2 2 3 2 2 3 + 1356 n p s - 4464 n p r - 3960 n p r s + 5700 n p r 2 2 2 2 4 2 3 2 5 + 3744 n p r s - 2736 n p r - 1032 n p r s + 378 n r 2 4 6 5 5 4 2 - 108 n r s - 42 n p - 846 n p r - 618 n p s + 1422 n p r 4 3 3 3 2 2 4 + 378 n p r s + 2148 n p r + 3204 n p r s - 5886 n p r 2 3 5 4 6 5 - 5700 n p r s + 4122 n p r + 3366 n p r s - 918 n r - 630 n r s 7 6 6 5 2 5 4 3 + p + 81 p r + 46 p s + 351 p r + 342 p r s - 1407 p r 4 2 3 4 3 3 2 5 2 4 - 1044 p r s + 951 p r + 324 p r s + 843 p r + 1182 p r s 6 5 7 6 3 3 3 2 - 1159 p r - 1146 p r s + 339 r + 296 r s + 419 n p - 647 n p r 3 2 3 2 3 3 3 3 2 + 610 n p s + 37 n p r - 1220 n p r s + 191 n r + 610 n r s 2 4 2 3 2 3 2 2 2 2 2 - 726 n p - 1074 n p r - 2721 n p s + 5748 n p r + 6333 n p r s 2 3 2 2 2 4 2 3 5 - 5370 n p r - 4503 n p r s + 1422 n r + 891 n r s + 240 n p 4 4 3 2 3 2 3 + 2334 n p r + 2082 n p s - 5037 n p r - 2886 n p r s + 537 n p r 2 2 4 3 5 4 - 2004 n p r s + 3501 n p r + 4338 n p r s - 1575 n r - 1530 n r s 6 5 5 4 2 4 3 3 - 14 p - 442 p r - 286 p s - 388 p r - 652 p r s + 3112 p r 3 2 2 4 2 3 5 4 6 + 2747 p r s - 2988 p r - 2079 p r s + 486 p r - 45 p r s + 234 r 5 3 2 3 3 3 2 3 + 315 r s - 416 n p + 442 n p r - 390 n p s - 26 n r + 390 n r s 2 3 2 2 2 2 2 2 2 + 1257 n p + 555 n p r + 3078 n p s - 3711 n p r - 4986 n p r s 2 3 2 2 4 3 3 + 1899 n r + 1908 n r s - 726 n p - 3588 n p r - 3978 n p s 2 2 2 3 2 4 + 7716 n p r + 5778 n p r s - 2934 n p r - 792 n p r s - 468 n r 3 5 4 4 3 2 3 - 1008 n r s + 80 p + 1262 p r + 936 p s - 613 p r + 234 p r s 2 3 2 2 4 3 5 4 - 3039 p r - 3240 p r s + 2859 p r + 2424 p r s - 549 r - 354 r s 3 3 3 2 2 2 2 + 220 n p - 120 n r + 100 n s - 1248 n p + 6 n p r - 1830 n p s 2 2 2 3 2 2 + 942 n r + 1530 n r s + 1257 n p + 3051 n p r + 4326 n p s 2 3 2 4 3 - 5553 n p r - 4992 n p r s + 1545 n r + 966 n r s - 242 p - 2034 p r 3 2 2 2 3 2 4 - 1745 p s + 1980 p r + 909 p r s + 1060 p r + 1587 p r s - 864 r 3 3 2 2 2 2 - 851 r s - 48 n + 660 n p - 72 n r + 444 n s - 1248 n p 2 3 2 - 1314 n p r - 2490 n p s + 1530 n r + 1602 n r s + 419 p + 1849 p r 2 2 3 2 2 + 1858 p s - 1805 p r - 1226 p r s + 29 r - 188 r s - 144 n + 660 n p 2 2 + 216 n r + 588 n s - 416 p - 878 p r - 1050 p s + 562 r + 462 r s / 3 \ |d | / - 144 n + 220 p + 168 r + 244 s - 48) |--- A[n](r, s)| / ( | 3 | / \dr / 3 2 2 3 2 2 (p - 3 p r + 3 p r - r - 10 p + 20 p r - 10 r + 33 p - 33 r - 36) %2) 3 5 3 5 3 4 2 3 4 3 3 3 + (n p r + n p s - 5 n p r - 5 n p r s + 10 n p r 3 3 2 3 2 4 3 2 3 3 5 3 4 + 10 n p r s - 10 n p r - 10 n p r s + 5 n p r + 5 n p r s 3 6 3 5 2 5 2 2 5 2 4 3 2 4 2 - n r - n r s - 3 n p r - 3 n p r s + 15 n p r + 15 n p r s 2 3 4 2 3 3 2 2 5 2 2 4 2 6 - 30 n p r - 30 n p r s + 30 n p r + 30 n p r s - 15 n p r 2 5 2 7 2 6 5 3 5 2 - 15 n p r s + 3 n r + 3 n r s + 3 n p r + 3 n p r s 4 4 4 3 3 5 3 4 2 6 - 15 n p r - 15 n p r s + 30 n p r + 30 n p r s - 30 n p r 2 5 7 6 8 7 5 4 - 30 n p r s + 15 n p r + 15 n p r s - 3 n r - 3 n r s - p r 5 3 4 5 4 4 3 6 3 5 2 7 - p r s + 5 p r + 5 p r s - 10 p r - 10 p r s + 10 p r 2 6 8 7 9 8 3 4 3 4 + 10 p r s - 5 p r - 5 p r s + r + r s - 7 n p r - 7 n p s 3 3 2 3 3 3 2 3 3 2 2 3 4 + 28 n p r + 28 n p r s - 42 n p r - 42 n p r s + 28 n p r 3 3 3 5 3 4 2 5 2 5 2 4 2 + 28 n p r s - 7 n r - 7 n r s + 3 n p r + 3 n p s + 6 n p r 2 4 2 3 3 2 3 2 2 2 4 2 2 3 + 6 n p r s - 54 n p r - 54 n p r s + 96 n p r + 96 n p r s 2 5 2 4 2 6 2 5 5 2 - 69 n p r - 69 n p r s + 18 n r + 18 n r s - 6 n p r 5 4 3 4 2 3 4 3 3 - 6 n p r s + 9 n p r + 9 n p r s + 24 n p r + 24 n p r s 2 5 2 4 6 5 7 - 66 n p r - 66 n p r s + 54 n p r + 54 n p r s - 15 n r 6 5 3 5 2 4 4 4 3 3 5 - 15 n r s + 3 p r + 3 p r s - 8 p r - 8 p r s + 2 p r 3 4 2 6 2 5 7 6 8 7 + 2 p r s + 12 p r + 12 p r s - 13 p r - 13 p r s + 4 r + 4 r s 3 3 3 3 3 2 2 3 2 3 3 + 19 n p r + 19 n p s - 57 n p r - 57 n p r s + 57 n p r 3 2 3 4 3 3 2 4 2 4 + 57 n p r s - 19 n r - 19 n r s - 21 n p r - 21 n p s 2 3 2 2 3 2 2 3 2 2 2 2 4 + 27 n p r + 27 n p r s + 45 n p r + 45 n p r s - 87 n p r 2 3 2 5 2 4 5 5 4 2 - 87 n p r s + 36 n r + 36 n r s + 3 n p r + 3 n p s + 27 n p r 4 3 3 3 2 2 4 2 3 + 27 n p r s - 81 n p r - 81 n p r s + 51 n p r + 51 n p r s 5 4 6 5 5 2 5 + 18 n p r + 18 n p r s - 18 n r - 18 n r s - 3 p r - 3 p r s 4 3 4 2 3 4 3 3 2 5 2 4 - 6 p r - 6 p r s + 35 p r + 35 p r s - 39 p r - 39 p r s 6 5 7 6 3 2 3 2 3 2 + 12 p r + 12 p r s + r + r s - 25 n p r - 25 n p s + 50 n p r 3 3 3 3 2 2 3 2 3 + 50 n p r s - 25 n r - 25 n r s + 57 n p r + 57 n p s 2 2 2 2 2 2 3 2 2 2 4 - 96 n p r - 96 n p r s + 21 n p r + 21 n p r s + 18 n r 2 3 4 4 3 2 3 + 18 n r s - 21 n p r - 21 n p s - 30 n p r - 30 n p r s 2 3 2 2 4 3 5 + 141 n p r + 141 n p r s - 108 n p r - 108 n p r s + 18 n r 4 5 5 4 2 4 3 3 3 2 + 18 n r s + p r + p s + 16 p r + 16 p r s - 17 p r - 17 p r s 2 4 2 3 5 4 6 5 - 30 p r - 30 p r s + 42 p r + 42 p r s - 12 r - 12 r s 3 3 3 2 3 2 2 2 2 + 16 n p r + 16 n p s - 16 n r - 16 n r s - 75 n p r - 75 n p s 2 2 2 2 3 2 2 3 + 102 n p r + 102 n p r s - 27 n r - 27 n r s + 57 n p r 3 2 2 2 3 2 4 + 57 n p s - 21 n p r - 21 n p r s - 81 n p r - 81 n p r s + 45 n r 3 4 4 3 2 3 2 3 + 45 n r s - 7 p r - 7 p s - 29 p r - 29 p r s + 54 p r 2 2 4 3 5 4 3 3 + 54 p r s - 9 p r - 9 p r s - 9 r - 9 r s - 4 n r - 4 n s 2 2 2 2 2 2 2 + 48 n p r + 48 n p s - 36 n r - 36 n r s - 75 n p r - 75 n p s 2 3 2 3 3 + 54 n p r + 54 n p r s + 9 n r + 9 n r s + 19 p r + 19 p s 2 2 2 3 2 4 3 2 + 18 p r + 18 p r s - 45 p r - 45 p r s + 12 r + 12 r s - 12 n r 2 2 2 2 - 12 n s + 48 n p r + 48 n p s - 24 n r - 24 n r s - 25 p r - 25 p s 2 3 2 + 2 p r + 2 p r s + 11 r + 11 r s - 12 n r - 12 n s + 16 p r + 16 p s / 4 \ 2 |d | 3 3 - 4 r - 4 r s - 4 r - 4 s) |--- A[n](r, s)|/(%1 (-4 + p - r)) - (n p | 4 | \dr / 3 2 3 2 3 3 2 4 2 3 2 2 2 - 3 n p r + 3 n p r - n r + 3 n p - 9 n p r + 9 n p r 2 3 5 4 3 2 2 3 6 5 - 3 n p r + 3 n p - 9 n p r + 9 n p r - 3 n p r + p - 3 p r 4 2 3 3 3 2 3 3 2 2 3 2 2 + 3 p r - p r - 6 n p + 12 n p r - 6 n r - 18 n p + 36 n p r 2 2 4 3 2 2 5 4 3 2 - 18 n p r - 18 n p + 36 n p r - 18 n p r - 6 p + 12 p r - 6 p r 3 3 2 2 2 3 2 4 + 12 n p - 12 n r + 36 n p - 36 n p r + 36 n p - 36 n p r + 12 p 3 3 2 2 3 - 12 p r - 8 n - 24 n p - 24 n p - 8 p ) A[n + 1](r, s)/(%2) /d \ 3 5 3 4 3 3 2 3 2 3 + |-- A[n + 1](r, s)| - (6 n p - 30 n p r + 60 n p r - 60 n p r \dr / 3 4 3 5 2 6 2 5 2 4 2 2 2 4 + 30 n p r - 6 n r + 9 n p - 36 n p r + 45 n p r - 45 n p r 2 5 2 6 7 6 5 2 4 3 + 36 n p r - 9 n r + 3 n p - 3 n p r - 27 n p r + 75 n p r 3 4 2 5 6 7 7 6 2 - 75 n p r + 27 n p r + 3 n p r - 3 n r + 3 p r - 12 p r 5 3 3 5 2 6 7 3 4 3 3 + 15 p r - 15 p r + 12 p r - 3 p r - 54 n p + 216 n p r 3 2 2 3 3 3 4 2 5 2 4 - 324 n p r + 216 n p r - 54 n r - 81 n p + 243 n p r 2 3 2 2 2 3 2 4 2 5 6 - 162 n p r - 162 n p r + 243 n p r - 81 n r - 21 n p 5 4 2 3 3 2 4 5 - 36 n p r + 333 n p r - 552 n p r + 333 n p r - 36 n p r 6 7 6 5 2 4 3 3 4 2 5 - 21 n r + 3 p - 42 p r + 108 p r - 69 p r - 69 p r + 108 p r 6 7 3 3 3 2 3 2 3 3 - 42 p r + 3 r + 189 n p - 567 n p r + 567 n p r - 189 n r 2 4 2 3 2 2 2 2 3 2 4 + 276 n p - 537 n p r - 45 n p r + 597 n p r - 291 n r 5 4 3 2 2 3 4 + 30 n p + 402 n p r - 1341 n p r + 1311 n p r - 357 n p r 5 6 5 4 2 3 3 2 4 5 - 45 n r - 30 p + 210 p r - 324 p r - 15 p r + 339 p r - 207 p r 6 3 2 3 3 2 2 3 2 2 + 27 r - 321 n p + 642 n p r - 321 n r - 429 n p + 324 n p r 2 2 2 3 4 3 2 2 + 639 n p r - 534 n r + 108 n p - 1290 n p r + 2259 n p r 3 4 5 4 3 2 2 3 - 1080 n p r + 3 n r + 117 p - 477 p r + 309 p r + 444 p r 4 5 3 3 2 2 2 - 492 p r + 99 r + 265 n p - 265 n r + 264 n p + 267 n p r 2 2 3 2 2 3 4 - 531 n r - 414 n p + 1770 n p r - 1503 n p r + 147 n r - 222 p 3 2 2 3 4 3 2 2 + 474 p r + 174 p r - 617 p r + 191 r - 85 n + 18 n p - 273 n r 2 2 3 2 2 3 + 516 n p - 996 n p r + 225 n r + 204 p - 96 p r - 402 p r + 209 r 2 2 2 - 57 n - 255 n p + 141 n r - 69 p - 117 p r + 129 r + 33 n - 8 p + 41 r / 2 \ |d | / 3 5 3 4 3 3 2 + 5) |--- A[n + 1](r, s)| / (4 n p - 20 n p r + 40 n p r | 2 | / \dr / 3 2 3 3 4 3 5 2 6 2 5 2 4 2 - 40 n p r + 20 n p r - 4 n r + 9 n p - 42 n p r + 75 n p r 2 3 3 2 2 4 2 5 2 6 7 6 - 60 n p r + 15 n p r + 6 n p r - 3 n r + 6 n p - 24 n p r 5 2 3 4 2 5 6 8 7 6 2 + 30 n p r - 30 n p r + 24 n p r - 6 n p r + p - 2 p r - 5 p r 5 3 4 4 3 5 2 6 3 4 3 3 + 20 p r - 25 p r + 14 p r - 3 p r - 42 n p + 168 n p r 3 2 2 3 3 3 4 2 5 2 4 - 252 n p r + 168 n p r - 42 n r - 96 n p + 354 n p r 2 3 2 2 2 3 2 4 2 5 6 - 456 n p r + 204 n p r + 24 n p r - 30 n r - 63 n p 5 4 2 3 3 2 4 5 + 186 n p r - 111 n p r - 156 n p r + 219 n p r - 78 n p r 6 7 5 2 4 3 3 4 2 5 6 + 3 n r - 9 p + 93 p r - 192 p r + 153 p r - 48 p r + 3 p r 3 3 3 2 3 2 3 3 2 4 + 172 n p - 516 n p r + 516 n p r - 172 n r + 399 n p 2 3 2 2 2 2 3 2 4 5 - 1080 n p r + 846 n p r - 48 n p r - 117 n r + 252 n p 4 3 2 2 3 4 5 6 - 462 n p r - 156 n p r + 720 n p r - 384 n p r + 30 n r + 24 p 5 4 2 3 3 2 4 5 6 3 2 + 108 p r - 501 p r + 616 p r - 282 p r + 36 p r - r - 345 n p 3 3 2 2 3 2 2 2 2 + 690 n p r - 345 n r - 807 n p + 1386 n p r - 351 n p r 2 3 4 3 2 2 3 4 - 228 n r - 462 n p + 234 n p r + 1035 n p r - 924 n p r + 117 n r 5 4 3 2 2 3 4 5 3 + 10 p - 512 p r + 1141 p r - 796 p r + 167 p r - 10 r + 342 n p 3 2 2 2 2 2 3 2 - 342 n r + 789 n p - 552 n p r - 237 n r + 324 n p + 606 n p r 2 3 4 3 2 2 3 - 1158 n p r + 228 n r - 162 p + 972 p r - 1155 p r + 384 p r 4 3 2 2 2 2 - 39 r - 135 n - 279 n p - 126 n r + 84 n p - 726 n p r + 237 n r 3 2 2 3 2 + 304 p - 828 p r + 465 p r - 76 r - 27 n - 180 n p + 126 n r 2 2 3 6 - 232 p + 284 p r - 79 r + 27 n + 69 p - 42 r - 9) + (4 n p 3 5 3 4 2 3 3 3 3 2 4 3 5 - 24 n p r + 60 n p r - 80 n p r + 60 n p r - 24 n p r 3 6 2 7 2 6 2 5 2 2 4 3 2 3 4 + 4 n r + 3 n p - 9 n p r - 9 n p r + 75 n p r - 135 n p r 2 2 5 2 6 2 7 7 6 2 5 3 + 117 n p r - 51 n p r + 9 n r + 6 n p r - 30 n p r + 54 n p r 4 4 3 5 2 6 7 8 7 2 - 30 n p r - 30 n p r + 54 n p r - 30 n p r + 6 n r + 3 p r 6 3 5 4 4 5 3 6 2 7 8 9 - 17 p r + 39 p r - 45 p r + 25 p r - 3 p r - 3 p r + r 3 5 3 4 3 3 2 3 2 3 3 4 - 46 n p + 230 n p r - 460 n p r + 460 n p r - 230 n p r 3 5 2 6 2 5 2 4 2 2 3 3 + 46 n r - 30 n p + 42 n p r + 240 n p r - 780 n p r 2 2 4 2 5 2 6 7 6 + 930 n p r - 510 n p r + 108 n r + 6 n p - 102 n p r 5 2 4 3 3 4 2 5 6 + 348 n p r - 420 n p r + 30 n p r + 354 n p r - 288 n p r 7 7 6 2 5 3 4 4 3 5 + 72 n r + 6 p r - 72 p r + 260 p r - 430 p r + 350 p r 2 6 7 8 3 4 3 3 3 2 2 - 116 p r - 8 p r + 10 r + 210 n p - 840 n p r + 1260 n p r 3 3 3 4 2 5 2 4 2 3 2 - 840 n p r + 210 n r + 102 n p + 120 n p r - 1500 n p r 2 2 3 2 4 2 5 6 5 + 2760 n p r - 2010 n p r + 528 n r - 72 n p + 636 n p r 4 2 3 3 2 4 5 6 7 - 1470 n p r + 960 n p r + 660 n p r - 1068 n p r + 354 n r + 3 p 6 5 2 4 3 3 4 2 5 6 - 93 p r + 597 p r - 1485 p r + 1725 p r - 903 p r + 123 p r 7 3 3 3 2 3 2 3 3 2 4 + 33 r - 488 n p + 1464 n p r - 1464 n p r + 488 n r - 96 n p 2 3 2 2 2 2 3 2 4 5 - 1080 n p r + 3816 n p r - 4008 n p r + 1368 n r + 342 n p 4 3 2 2 3 4 5 - 1902 n p r + 2724 n p r - 180 n p r - 1914 n p r + 930 n r 6 5 4 2 3 3 2 4 5 - 38 p + 570 p r - 2376 p r + 4076 p r - 3102 p r + 858 p r 6 3 2 3 3 2 2 3 2 2 + 12 r + 610 n p - 1220 n p r + 610 n r - 207 n p + 2451 n p r 2 2 2 3 4 3 2 2 - 4281 n p r + 2037 n r - 822 n p + 2874 n p r - 1860 n p r 3 4 5 4 3 2 2 3 - 1614 n p r + 1422 n r + 194 p - 1792 p r + 5021 p r - 5641 p r 4 5 3 3 2 2 2 + 2417 p r - 199 r - 390 n p + 390 n r + 582 n p - 2334 n p r 2 2 3 2 2 3 4 + 1752 n r + 1050 n p - 1986 n p r - 348 n p r + 1284 n r - 516 p 3 2 2 3 4 3 2 + 3114 p r - 5664 p r + 3660 p r - 594 r + 100 n - 510 n p 2 2 2 3 2 + 810 n r - 666 n p + 312 n p r + 654 n r + 769 p - 2973 p r 2 3 2 2 + 3129 p r - 825 r + 156 n + 150 n p + 162 n r - 638 p + 1426 p r / 3 \ 2 |d | / - 632 r + 12 n + 270 p - 258 r - 44) |--- A[n + 1](r, s)| / ( | 3 | / \dr / 3 5 3 4 3 3 2 3 2 3 3 4 (-4 + p - r) (4 n p - 20 n p r + 40 n p r - 40 n p r + 20 n p r 3 5 2 6 2 5 2 4 2 2 3 3 2 2 4 - 4 n r + 9 n p - 42 n p r + 75 n p r - 60 n p r + 15 n p r 2 5 2 6 7 6 5 2 3 4 + 6 n p r - 3 n r + 6 n p - 24 n p r + 30 n p r - 30 n p r 2 5 6 8 7 6 2 5 3 4 4 + 24 n p r - 6 n p r + p - 2 p r - 5 p r + 20 p r - 25 p r 3 5 2 6 3 4 3 3 3 2 2 3 3 + 14 p r - 3 p r - 42 n p + 168 n p r - 252 n p r + 168 n p r 3 4 2 5 2 4 2 3 2 2 2 3 - 42 n r - 96 n p + 354 n p r - 456 n p r + 204 n p r 2 4 2 5 6 5 4 2 3 3 + 24 n p r - 30 n r - 63 n p + 186 n p r - 111 n p r - 156 n p r 2 4 5 6 7 5 2 4 3 + 219 n p r - 78 n p r + 3 n r - 9 p + 93 p r - 192 p r 3 4 2 5 6 3 3 3 2 3 2 + 153 p r - 48 p r + 3 p r + 172 n p - 516 n p r + 516 n p r 3 3 2 4 2 3 2 2 2 2 3 - 172 n r + 399 n p - 1080 n p r + 846 n p r - 48 n p r 2 4 5 4 3 2 2 3 - 117 n r + 252 n p - 462 n p r - 156 n p r + 720 n p r 4 5 6 5 4 2 3 3 - 384 n p r + 30 n r + 24 p + 108 p r - 501 p r + 616 p r 2 4 5 6 3 2 3 3 2 - 282 p r + 36 p r - r - 345 n p + 690 n p r - 345 n r 2 3 2 2 2 2 2 3 4 - 807 n p + 1386 n p r - 351 n p r - 228 n r - 462 n p 3 2 2 3 4 5 4 + 234 n p r + 1035 n p r - 924 n p r + 117 n r + 10 p - 512 p r 3 2 2 3 4 5 3 3 + 1141 p r - 796 p r + 167 p r - 10 r + 342 n p - 342 n r 2 2 2 2 2 3 2 2 + 789 n p - 552 n p r - 237 n r + 324 n p + 606 n p r - 1158 n p r 3 4 3 2 2 3 4 3 + 228 n r - 162 p + 972 p r - 1155 p r + 384 p r - 39 r - 135 n 2 2 2 2 3 2 - 279 n p - 126 n r + 84 n p - 726 n p r + 237 n r + 304 p - 828 p r 2 3 2 2 2 + 465 p r - 76 r - 27 n - 180 n p + 126 n r - 232 p + 284 p r - 79 r 3 5 3 4 3 3 2 + 27 n + 69 p - 42 r - 9)) - (n p - 5 n p r + 10 n p r 3 2 3 3 4 3 5 2 5 2 4 2 2 3 3 - 10 n p r + 5 n p r - n r + 3 n p r - 15 n p r + 30 n p r 2 2 4 2 5 2 6 5 2 4 3 3 4 - 30 n p r + 15 n p r - 3 n r + 3 n p r - 15 n p r + 30 n p r 2 5 6 7 5 3 4 4 3 5 2 6 - 30 n p r + 15 n p r - 3 n r + p r - 5 p r + 10 p r - 10 p r 7 8 3 4 3 3 3 2 2 3 3 3 4 + 5 p r - r - 7 n p + 28 n p r - 42 n p r + 28 n p r - 7 n r 2 5 2 4 2 3 2 2 2 3 2 4 + 3 n p - 36 n p r + 114 n p r - 156 n p r + 99 n p r 2 5 5 4 2 3 3 2 4 - 24 n r + 6 n p r - 51 n p r + 144 n p r - 186 n p r 5 6 5 2 4 3 3 4 2 5 + 114 n p r - 27 n r + 3 p r - 22 p r + 58 p r - 72 p r 6 7 3 3 3 2 3 2 3 3 + 43 p r - 10 r + 19 n p - 57 n p r + 57 n p r - 19 n r 2 4 2 3 2 2 2 2 3 2 4 5 - 21 n p + 141 n p r - 297 n p r + 255 n p r - 78 n r + 3 n p 4 3 2 2 3 4 5 5 - 57 n p r + 255 n p r - 453 n p r + 354 n p r - 102 n r + 3 p r 4 2 3 3 2 4 5 6 3 2 - 36 p r + 133 p r - 213 p r + 156 p r - 43 r - 25 n p 3 3 2 2 3 2 2 2 2 2 3 + 50 n p r - 25 n r + 57 n p - 246 n p r + 321 n p r - 132 n r 4 3 2 2 3 4 5 - 21 n p + 198 n p r - 543 n p r + 576 n p r - 210 n r + p 4 3 2 2 3 4 5 3 3 - 26 p r + 151 p r - 332 p r + 310 p r - 104 r + 16 n p - 16 n r 2 2 2 2 2 3 2 2 - 75 n p + 198 n p r - 123 n r + 57 n p - 321 n p r + 519 n p r 3 4 3 2 2 3 4 3 - 255 n r - 7 p + 85 p r - 288 p r + 365 p r - 155 r - 4 n 2 2 2 2 3 2 + 48 n p - 60 n r - 75 n p + 246 n p r - 183 n r + 19 p - 132 p r 2 3 2 2 2 + 255 p r - 146 r - 12 n + 48 n p - 72 n r - 25 p + 98 p r - 85 r / 4 \ |d | - 12 n + 16 p - 28 r - 4) |--- A[n + 1](r, s)|/(%1 (-4 + p - r)) = 0 | 4 | \dr / 3 4 3 3 3 2 2 3 3 3 4 2 5 %1 := 4 n p - 16 n p r + 24 n p r - 16 n p r + 4 n r + 9 n p 2 4 2 3 2 2 2 3 2 4 2 5 6 - 33 n p r + 42 n p r - 18 n p r - 3 n p r + 3 n r + 6 n p 5 4 2 3 3 2 4 5 7 6 - 18 n p r + 12 n p r + 12 n p r - 18 n p r + 6 n p r + p - p r 5 2 4 3 3 4 2 5 3 3 3 2 - 6 p r + 14 p r - 11 p r + 3 p r - 30 n p + 90 n p r 3 2 3 3 2 4 2 3 2 2 2 - 90 n p r + 30 n r - 69 n p + 186 n p r - 144 n p r 2 3 2 4 5 4 3 2 2 3 + 6 n p r + 21 n r - 45 n p + 87 n p r + 12 n p r - 108 n p r 4 5 6 5 4 2 3 3 2 4 + 57 n p r - 3 n r - 6 p - 9 p r + 66 p r - 84 p r + 36 p r 5 3 2 3 3 2 2 3 2 2 - 3 p r + 82 n p - 164 n p r + 82 n r + 192 n p - 330 n p r 2 2 2 3 4 3 2 2 3 + 84 n p r + 54 n r + 117 n p - 84 n p r - 204 n p r + 192 n p r 4 5 4 3 2 2 3 4 5 - 21 n r + 6 p + 87 p r - 216 p r + 148 p r - 26 p r + r 3 3 2 2 2 2 2 3 - 99 n p + 99 n r - 231 n p + 165 n p r + 66 n r - 111 n p 2 2 3 4 3 2 2 - 129 n p r + 294 n p r - 54 n r + 28 p - 223 p r + 270 p r 3 4 3 2 2 2 - 82 p r + 7 r + 45 n + 96 n p + 39 n r - 9 n p + 210 n p r 2 3 2 2 3 2 - 66 n r - 78 p + 225 p r - 120 p r + 18 r + 9 n + 57 n p - 39 n r 2 2 + 70 p - 83 p r + 22 r - 9 n - 22 p + 13 r + 3 3 3 3 2 3 2 3 3 2 4 2 3 %2 := 4 n p - 12 n p r + 12 n p r - 4 n r + 9 n p - 24 n p r 2 2 2 2 4 5 4 2 3 4 6 + 18 n p r - 3 n r + 6 n p - 12 n p r + 12 n p r - 6 n p r + p 4 2 3 3 2 4 3 2 3 3 2 2 3 - 6 p r + 8 p r - 3 p r - 18 n p + 36 n p r - 18 n r - 42 n p 2 2 2 2 2 3 4 3 2 2 + 72 n p r - 18 n p r - 12 n r - 27 n p + 24 n p r + 36 n p r 3 4 5 4 3 2 2 3 4 - 36 n p r + 3 n r - 3 p - 12 p r + 36 p r - 24 p r + 3 p r 3 3 2 2 2 2 2 3 + 28 n p - 28 n r + 66 n p - 48 n p r - 18 n r + 36 n p 2 2 3 4 3 2 2 3 + 24 n p r - 72 n p r + 12 n r - 3 p + 48 p r - 60 p r + 16 p r 4 3 2 2 2 2 3 - r - 15 n - 33 n p - 12 n r - 3 n p - 60 n p r + 18 n r + 19 p 2 2 3 2 2 - 60 p r + 30 p r - 4 r - 3 n - 18 n p + 12 n r - 21 p + 24 p r 2 - 6 r + 3 n + 7 p - 4 r - 1 6 5 5 4 2 4 4 2 3 3 (8 n + 36 n q - 24 n s + 66 n q - 84 n q s + 24 n s + 63 n q 3 2 3 2 3 3 2 4 2 3 2 2 2 - 114 n q s + 60 n q s - 8 n s + 33 n q - 75 n q s + 54 n q s 2 3 5 4 3 2 2 3 6 5 - 12 n q s + 9 n q - 24 n q s + 21 n q s - 6 n q s + q - 3 q s 4 2 3 3 5 4 4 3 2 3 3 2 + 3 q s - q s - 4 n - 8 n q - 4 n s - n q - 28 n q s + 20 n s 2 3 2 2 2 2 2 3 4 3 + 7 n q - 45 n q s + 48 n q s - 12 n s + 5 n q - 26 n q s 2 2 3 5 4 3 2 2 3 4 + 33 n q s - 12 n q s + q - 5 q s + 7 q s - 3 q s - 26 n 3 3 2 2 2 2 2 3 2 - 71 n q + 38 n s - 69 n q + 63 n q s - 6 n s - 28 n q + 30 n q s 2 3 4 3 2 2 3 3 2 + 3 n q s - 6 n s - 4 q + 4 q s + 3 q s - 3 q s - 7 n - 27 n q 2 2 2 3 2 2 3 + 33 n s - 24 n q + 42 n q s - 9 n s - 6 q + 12 q s - 3 q s - s 2 2 2 + 16 n + 11 n q + 10 n s + q + 7 q s - 2 s + 11 n + 5 q + s + 2) / 2 3 7 6 6 A[n](r, s) / ((n + q - s) (n - s + 1) ) - (12 n + 60 n q - 36 n s / 5 2 5 5 2 4 3 4 2 4 2 + 123 n q - 132 n q s + 24 n s + 132 n q - 177 n q s + 24 n q s 4 3 3 4 3 3 3 2 2 3 3 3 4 + 24 n s + 78 n q - 96 n q s - 66 n q s + 120 n q s - 36 n s 2 5 2 4 2 3 2 2 2 3 2 4 + 24 n q - 6 n q s - 120 n q s + 174 n q s - 84 n q s 2 5 6 5 4 2 3 3 2 4 + 12 n s + 3 n q + 12 n q s - 66 n q s + 96 n q s - 57 n q s 5 6 5 2 4 3 3 4 2 5 6 + 12 n q s + 3 q s - 12 q s + 18 q s - 12 q s + 3 q s - 6 n 5 5 4 2 4 4 2 3 3 - 9 n q - 18 n s + 21 n q - 129 n q s + 84 n s + 60 n q 3 2 3 2 3 3 2 4 2 3 - 276 n q s + 294 n q s - 84 n s + 54 n q - 252 n q s 2 2 2 2 3 2 4 5 4 + 342 n q s - 162 n q s + 18 n s + 21 n q - 102 n q s 3 2 2 3 4 5 6 5 4 2 + 156 n q s - 84 n q s + 3 n q s + 6 n s + 3 q - 15 q s + 24 q s 3 3 2 4 5 5 4 4 3 2 - 12 q s - 3 q s + 3 q s - 41 n - 145 n q + 85 n s - 194 n q 3 3 2 2 3 2 2 2 2 2 3 + 196 n q s - 26 n s - 120 n q + 138 n q s + 18 n q s - 38 n s 4 3 2 2 3 4 5 4 - 33 n q + 24 n q s + 66 n q s - 76 n q s + 19 n s - 3 q - 3 q s 3 2 2 3 4 5 4 3 3 2 2 + 24 q s - 26 q s + 7 q s + s - 4 n - 40 n q + 64 n s - 72 n q 2 2 2 3 2 2 3 4 + 168 n q s - 72 n s - 45 n q + 126 n q s - 84 n q s + 8 n s - 9 q 3 2 2 3 4 3 2 2 2 + 27 q s - 18 q s - 4 q s + 4 s + 28 n + 54 n q - 24 n s + 27 n q 2 3 2 2 3 2 2 - 24 n s + 3 q + 9 q s - 18 q s + 4 s + 4 n + 17 n q - 26 n s + 7 q 2 /d \ / - 11 q s - 2 s - 7 n - q - 5 s - 2) |-- A[n](r, s)| / ((n + q - s) \ds / / 2 2 2 3 6 (n + 2 n q - 2 n s + q - 2 q s + s - n - q + s) (n - s + 1) ) + (6 n 5 5 4 2 4 4 2 3 3 + 21 n q - 6 n s + 27 n q - 3 n q s - 12 n s + 15 n q 3 2 3 2 3 3 2 4 2 3 2 2 2 + 18 n q s - 42 n q s + 12 n s + 3 n q + 21 n q s - 36 n q s 2 3 2 4 4 3 2 2 3 4 + 6 n q s + 6 n s + 6 n q s - 3 n q s - 18 n q s + 21 n q s 5 4 2 3 3 2 4 5 5 4 4 - 6 n s + 3 q s - 9 q s + 9 q s - 3 q s + 6 n + 27 n q - 24 n s 3 2 3 3 2 2 3 2 2 2 2 + 42 n q - 60 n q s + 12 n s + 27 n q - 36 n q s - 18 n q s 2 3 4 3 2 2 3 4 4 + 24 n s + 6 n q + 6 n q s - 54 n q s + 60 n q s - 18 n s + 6 q s 3 2 2 3 4 4 3 3 2 2 - 21 q s + 24 q s - 9 q s - 16 n - 25 n q - 14 n s - 3 n q 2 2 2 3 2 2 3 4 - 63 n q s + 42 n s + 9 n q - 60 n q s + 57 n q s - 10 n s + 3 q 3 2 2 3 4 3 2 2 2 - 15 q s + 15 q s - q s - 2 s - 22 n - 45 n q + 24 n s - 24 n q 2 3 2 2 3 + 6 n q s + 18 n s - 3 q - 6 q s + 15 q s - 4 s - 12 n q + 24 n s / 2 \ 2 |d | / - 6 q + 12 q s + 8 n + 2 q + 4 s + 2) |--- A[n](r, s)| / ( | 2 | / \ds / 2 2 2 3 4 (n + 2 n q - 2 n s + q - 2 q s + s - n - q + s) (n - s + 1) ) - (n 3 3 2 2 3 3 4 3 2 + n q + 2 n s + 3 n q s + 3 n q s - 2 n s + q s - s + 4 n + 3 n q 2 2 3 2 + 6 n s + 6 n q s + 3 q s - 2 s + 6 n + 3 n q + 6 n s + 3 q s + 4 n / 3 \ |d | / + q + 2 s + 1) |--- A[n](r, s)| / ( | 3 | / \ds / 3 2 2 3 2 2 3 n - 3 n s + 3 n s - s + 3 n - 6 n s + 3 s + 3 n - 3 s + 1) - (n 2 2 2 2 3 2 2 3 + 3 n q - 3 n s + 3 n q - 6 n q s + 3 n s + q - 3 q s + 3 q s - s 2 2 2 3 - 3 n - 6 n q + 6 n s - 3 q + 6 q s - 3 s + 2 n + 2 q - 2 s) q / 3 3 6 2 5 3 A[n + 1](r, s) / ((n + q - s + 1) (n - s + 1) ) - (3 n q + 14 n q / 5 2 4 4 4 3 4 2 2 3 5 - 18 n q s + 25 n q - 70 n q s + 45 n q s + 20 n q 3 4 3 3 2 3 2 3 2 6 2 5 - 100 n q s + 140 n q s - 60 n q s + 5 n q - 60 n q s 2 4 2 2 3 3 2 2 4 7 6 + 150 n q s - 140 n q s + 45 n q s - 2 n q - 10 n q s 5 2 4 3 3 4 2 5 8 7 + 60 n q s - 100 n q s + 70 n q s - 18 n q s - q + 2 q s 6 2 5 3 4 4 3 5 2 6 6 5 2 + 5 q s - 20 q s + 25 q s - 14 q s + 3 q s - 3 n q - 15 n q 5 4 3 4 2 4 2 3 4 3 3 + 18 n q s - 24 n q + 75 n q s - 45 n q s - 6 n q + 96 n q s 3 2 2 3 3 2 5 2 4 2 3 2 - 150 n q s + 60 n q s + 21 n q + 18 n q s - 144 n q s 2 2 3 2 4 6 5 4 2 + 150 n q s - 45 n q s + 21 n q - 42 n q s - 18 n q s 3 3 2 4 5 7 6 5 2 + 96 n q s - 75 n q s + 18 n q s + 6 q - 21 q s + 21 q s 4 3 3 4 2 5 6 6 5 5 4 2 + 6 q s - 24 q s + 15 q s - 3 q s + n + 3 n q - 6 n s - 12 n q 4 4 2 3 3 3 2 3 2 3 3 - 15 n q s + 15 n s - 52 n q + 48 n q s + 30 n q s - 20 n s 2 4 2 3 2 2 2 2 3 2 4 5 - 69 n q + 156 n q s - 72 n q s - 30 n q s + 15 n s - 39 n q 4 3 2 2 3 4 5 6 + 138 n q s - 156 n q s + 48 n q s + 15 n q s - 6 n s - 8 q 5 4 2 3 3 2 4 5 6 5 4 + 39 q s - 69 q s + 52 q s - 12 q s - 3 q s + s + n + 17 n q 4 3 2 3 3 2 2 3 2 2 - 5 n s + 40 n q - 68 n q s + 10 n s + 27 n q - 120 n q s 2 2 2 3 4 3 2 2 3 + 102 n q s - 10 n s - 3 n q - 54 n q s + 120 n q s - 68 n q s 4 5 4 3 2 2 3 4 5 4 + 5 n s - 6 q + 3 q s + 27 q s - 40 q s + 17 q s - s - 4 n 3 3 2 2 2 2 2 3 2 + 2 n q + 16 n s + 33 n q - 6 n q s - 24 n s + 39 n q - 66 n q s 2 3 4 3 2 2 3 4 3 + 6 n q s + 16 n s + 12 q - 39 q s + 33 q s - 2 q s - 4 s - 6 n 2 2 2 2 3 2 - 21 n q + 18 n s - 9 n q + 42 n q s - 18 n s + 4 q + 9 q s 2 3 2 2 2 - 21 q s + 6 s + n - 13 n q - 2 n s - 8 q + 13 q s + s + 5 n - q /d \ / - 5 s + 2) |-- A[n + 1](r, s)| / ((n + q - s + 1) \ds / / 2 2 2 2 3 6 (n + 2 n q - 2 n s + q - 2 q s + s + n + q - s) (n - s + 1) ) - (3 n q 5 2 5 4 3 4 2 4 2 3 3 + 12 n q - 18 n q s + 15 n q - 60 n q s + 45 n q s - 60 n q s 3 2 2 3 3 2 5 2 3 2 2 2 3 + 120 n q s - 60 n q s - 15 n q + 90 n q s - 120 n q s 2 4 6 5 3 3 2 4 5 + 45 n q s - 12 n q + 30 n q s - 60 n q s + 60 n q s - 18 n q s 7 6 5 2 3 4 2 5 6 6 - 3 q + 12 q s - 15 q s + 15 q s - 12 q s + 3 q s - 3 n 5 5 4 2 4 4 2 3 3 - 12 n q + 18 n s - 3 n q + 60 n q s - 45 n s + 48 n q 3 2 3 2 3 3 2 4 2 3 + 12 n q s - 120 n q s + 60 n s + 87 n q - 144 n q s 2 2 2 2 3 2 4 5 4 - 18 n q s + 120 n q s - 45 n s + 60 n q - 174 n q s 3 2 2 3 4 5 6 5 + 144 n q s + 12 n q s - 60 n q s + 18 n s + 15 q - 60 q s 4 2 3 3 2 4 5 6 5 4 + 87 q s - 48 q s - 3 q s + 12 q s - 3 s - 3 n - 33 n q 4 3 2 3 3 2 2 3 2 2 + 15 n s - 99 n q + 132 n q s - 30 n s - 129 n q + 297 n q s 2 2 2 3 4 3 2 2 3 - 198 n q s + 30 n s - 78 n q + 258 n q s - 297 n q s + 132 n q s 4 5 4 3 2 2 3 4 5 - 15 n s - 18 q + 78 q s - 129 q s + 99 q s - 33 q s + 3 s 4 3 3 2 2 2 2 2 3 + 12 n + 30 n q - 48 n s + 12 n q - 90 n q s + 72 n s - 18 n q 2 2 3 4 3 2 2 3 - 24 n q s + 90 n q s - 48 n s - 12 q + 18 q s + 12 q s - 30 q s 4 3 2 2 2 2 + 12 s + 14 n + 53 n q - 42 n s + 63 n q - 106 n q s + 42 n s 3 2 2 3 2 2 + 24 q - 63 q s + 53 q s - 14 s - 3 n - 2 n q + 6 n s + 3 q + 2 q s / 2 \ 2 |d | / - 3 s - 3 n - 7 q + 3 s + 2) |--- A[n + 1](r, s)| / ( | 2 | / \ds / 2 2 2 2 3 (n + 2 n q - 2 n s + q - 2 q s + s + n + q - s) (n - s + 1) (n 2 2 2 2 3 2 2 3 + 3 n q - 3 n s + 3 n q - 6 n q s + 3 n s + q - 3 q s + 3 q s - s 4 3 2 2 2 2 3 2 - n - q + s)) - (n - 4 n s - 6 n q + 6 n s - 8 n q + 12 n q s 3 4 3 2 2 4 3 2 2 - 4 n s - 3 q + 8 q s - 6 q s + s + 4 n + 18 n q - 12 n s 2 2 3 2 2 3 2 + 24 n q - 36 n q s + 12 n s + 10 q - 24 q s + 18 q s - 4 s - 5 n 2 2 - 8 n q + 10 n s - 3 q + 8 q s - 5 s - 6 n - 8 q + 6 s + 2) / 3 \ |d | / 3 2 2 2 2 |--- A[n + 1](r, s)| / ((n + 3 n q - 3 n s + 3 n q - 6 n q s + 3 n s | 3 | / \ds / 3 2 2 3 + q - 3 q s + 3 q s - s - n - q + s) (n - s + 1)) / 4 \ |d | + |--- A[n + 1](r, s)| = 0 | 4 | \ds / and in Maple notation (n^3*p^4-3*n^3*p^3*r+n^3*p^3*s+3*n^3*p^2*r^2-3*n^3*p^2*r*s-n^3*p*r^3+3*n^3*p*r^ 2*s-n^3*r^3*s-3*n^2*p^5+9*n^2*p^4*r-3*n^2*p^4*s-9*n^2*p^3*r^2+9*n^2*p^3*r*s+3*n ^2*p^2*r^3-9*n^2*p^2*r^2*s+3*n^2*p*r^3*s+3*n*p^6-9*n*p^5*r+3*n*p^5*s+9*n*p^4*r^ 2-9*n*p^4*r*s-3*n*p^3*r^3+9*n*p^3*r^2*s-3*n*p^2*r^3*s-p^7+3*p^6*r-p^6*s-3*p^5*r ^2+3*p^5*r*s+p^4*r^3-3*p^4*r^2*s+p^3*r^3*s-7*n^3*p^3+15*n^3*p^2*r-6*n^3*p^2*s-9 *n^3*p*r^2+12*n^3*p*r*s+n^3*r^3-6*n^3*r^2*s+27*n^2*p^4-63*n^2*p^3*r+24*n^2*p^3* s+45*n^2*p^2*r^2-54*n^2*p^2*r*s-9*n^2*p*r^3+36*n^2*p*r^2*s-6*n^2*r^3*s-33*n*p^5 +81*n*p^4*r-30*n*p^4*s-63*n*p^3*r^2+72*n*p^3*r*s+15*n*p^2*r^3-54*n*p^2*r^2*s+12 *n*p*r^3*s+13*p^6-33*p^5*r+12*p^5*s+27*p^4*r^2-30*p^4*r*s-7*p^3*r^3+24*p^3*r^2* s-6*p^2*r^3*s+18*n^3*p^2-24*n^3*p*r+12*n^3*p*s+6*n^3*r^2-12*n^3*r*s-96*n^2*p^3+ 162*n^2*p^2*r-72*n^2*p^2*s-72*n^2*p*r^2+108*n^2*p*r*s+6*n^2*r^3-36*n^2*r^2*s+ 150*n*p^4-288*n*p^3*r+120*n*p^3*s+162*n*p^2*r^2-216*n*p^2*r*s-24*n*p*r^3+108*n* p*r^2*s-12*n*r^3*s-72*p^5+150*p^4*r-60*p^4*s-96*p^3*r^2+120*p^3*r*s+18*p^2*r^3-\ 72*p^2*r^2*s+12*p*r^3*s-20*n^3*p+12*n^3*r-8*n^3*s+168*n^2*p^2-180*n^2*p*r+96*n^ 2*p*s+36*n^2*r^2-72*n^2*r*s-360*n*p^3+504*n*p^2*r-240*n*p^2*s-180*n*p*r^2+288*n *p*r*s+12*n*r^3-72*n*r^2*s+220*p^4-360*p^3*r+160*p^3*s+168*p^2*r^2-240*p^2*r*s-\ 20*p*r^3+96*p*r^2*s-8*r^3*s+8*n^3-144*n^2*p+72*n^2*r-48*n^2*s+480*n*p^2-432*n*p *r+240*n*p*s+72*n*r^2-144*n*r*s-400*p^3+480*p^2*r-240*p^2*s-144*p*r^2+240*p*r*s +8*r^3-48*r^2*s+48*n^2-336*n*p+144*n*r-96*n*s+432*p^2-336*p*r+192*p*s+48*r^2-96 *r*s+96*n-256*p+96*r-64*s+64)/(4*n^3*p^3-12*n^3*p^2*r+12*n^3*p*r^2-4*n^3*r^3+9* n^2*p^4-24*n^2*p^3*r+18*n^2*p^2*r^2-3*n^2*r^4+6*n*p^5-12*n*p^4*r+12*n*p^2*r^3-6 *n*p*r^4+p^6-6*p^4*r^2+8*p^3*r^3-3*p^2*r^4-18*n^3*p^2+36*n^3*p*r-18*n^3*r^2-42* n^2*p^3+72*n^2*p^2*r-18*n^2*p*r^2-12*n^2*r^3-27*n*p^4+24*n*p^3*r+36*n*p^2*r^2-\ 36*n*p*r^3+3*n*r^4-3*p^5-12*p^4*r+36*p^3*r^2-24*p^2*r^3+3*p*r^4+28*n^3*p-28*n^3 *r+66*n^2*p^2-48*n^2*p*r-18*n^2*r^2+36*n*p^3+24*n*p^2*r-72*n*p*r^2+12*n*r^3-3*p ^4+48*p^3*r-60*p^2*r^2+16*p*r^3-r^4-15*n^3-33*n^2*p-12*n^2*r-3*n*p^2-60*n*p*r+ 18*n*r^2+19*p^3-60*p^2*r+30*p*r^2-4*r^3-3*n^2-18*n*p+12*n*r-21*p^2+24*p*r-6*r^2 +3*n+7*p-4*r-1)*A[n](r,s)-(3*n^3*p^4-8*n^3*p^3*r+4*n^3*p^3*s+6*n^3*p^2*r^2-12*n ^3*p^2*r*s+12*n^3*p*r^2*s-n^3*r^4-4*n^3*r^3*s-6*n^2*p^5+12*n^2*p^4*r-9*n^2*p^4* s+24*n^2*p^3*r*s-12*n^2*p^2*r^3-18*n^2*p^2*r^2*s+6*n^2*p*r^4+3*n^2*r^4*s+3*n*p^ 6+6*n*p^5*s-18*n*p^4*r^2-12*n*p^4*r*s+24*n*p^3*r^3-9*n*p^2*r^4+12*n*p^2*r^3*s-6 *n*p*r^4*s-4*p^6*r-p^6*s+12*p^5*r^2-12*p^4*r^3+6*p^4*r^2*s+4*p^3*r^4-8*p^3*r^3* s+3*p^2*r^4*s-18*n^3*p^3+36*n^3*p^2*r-18*n^3*p^2*s-18*n^3*p*r^2+36*n^3*p*r*s-18 *n^3*r^2*s+54*n^2*p^4-96*n^2*p^3*r+66*n^2*p^3*s+18*n^2*p^2*r^2-144*n^2*p^2*r*s+ 36*n^2*p*r^3+90*n^2*p*r^2*s-12*n^2*r^4-12*n^2*r^3*s-39*n*p^5+24*n*p^4*r-63*n*p^ 4*s+108*n*p^3*r^2+120*n*p^3*r*s-132*n*p^2*r^3-36*n*p^2*r^2*s+39*n*p*r^4-36*n*p* r^3*s+15*n*r^4*s+3*p^6+36*p^5*r+15*p^5*s-108*p^4*r^2-12*p^4*r*s+96*p^3*r^3-36*p ^3*r^2*s-27*p^2*r^4+48*p^2*r^3*s-15*p*r^4*s+40*n^3*p^2-52*n^3*p*r+28*n^3*p*s+12 *n^3*r^2-28*n^3*r*s-186*n^2*p^3+264*n^2*p^2*r-174*n^2*p^2*s-54*n^2*p*r^2+264*n^ 2*p*r*s-24*n^2*r^3-90*n^2*r^2*s+198*n*p^4-168*n*p^3*r+252*n*p^3*s-216*n*p^2*r^2 -408*n*p^2*r*s+228*n*p*r^3+144*n*p*r^2*s-42*n*r^4+12*n*r^3*s-33*p^5-120*p^4*r-\ 87*p^4*s+372*p^3*r^2+96*p^3*r*s-280*p^2*r^3+60*p^2*r^2*s+61*p*r^4-88*p*r^3*s+19 *r^4*s-39*n^3*p+24*n^3*r-15*n^3*s+309*n^2*p^2-300*n^2*p*r+201*n^2*p*s+36*n^2*r^ 2-156*n^2*r*s-507*n*p^3+420*n*p^2*r-483*n*p^2*s+162*n*p*r^2+564*n*p*r*s-120*n*r ^3-126*n*r^2*s+147*p^4+172*p^3*r+253*p^3*s-606*p^2*r^2-276*p^2*r*s+348*p*r^3-6* p*r^2*s-46*r^4+44*r^3*s+14*n^3-249*n^2*p+120*n^2*r-87*n^2*s+696*n*p^2-444*n*p*r +450*n*p*s-36*n*r^2-276*n*r*s-339*p^3-72*p^2*r-393*p^2*s+462*p*r^2+336*p*r*s-\ 152*r^3-30*r^2*s+78*n^2-489*n*p+168*n*r-165*n*s+427*p^2-52*p*r+313*p*s-132*r^2-\ 148*r*s+138*n-279*p+40*r-101*s+74)/(4*n^3*p^3-12*n^3*p^2*r+12*n^3*p*r^2-4*n^3*r ^3+9*n^2*p^4-24*n^2*p^3*r+18*n^2*p^2*r^2-3*n^2*r^4+6*n*p^5-12*n*p^4*r+12*n*p^2* 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117*n^2*r^4+252*n*p^5-462*n*p^4*r-156*n*p^3*r^2+720*n*p^2*r^3-384*n*p*r^4+30*n* r^5+24*p^6+108*p^5*r-501*p^4*r^2+616*p^3*r^3-282*p^2*r^4+36*p*r^5-r^6-345*n^3*p ^2+690*n^3*p*r-345*n^3*r^2-807*n^2*p^3+1386*n^2*p^2*r-351*n^2*p*r^2-228*n^2*r^3 -462*n*p^4+234*n*p^3*r+1035*n*p^2*r^2-924*n*p*r^3+117*n*r^4+10*p^5-512*p^4*r+ 1141*p^3*r^2-796*p^2*r^3+167*p*r^4-10*r^5+342*n^3*p-342*n^3*r+789*n^2*p^2-552*n ^2*p*r-237*n^2*r^2+324*n*p^3+606*n*p^2*r-1158*n*p*r^2+228*n*r^3-162*p^4+972*p^3 *r-1155*p^2*r^2+384*p*r^3-39*r^4-135*n^3-279*n^2*p-126*n^2*r+84*n*p^2-726*n*p*r +237*n*r^2+304*p^3-828*p^2*r+465*p*r^2-76*r^3-27*n^2-180*n*p+126*n*r-232*p^2+ 284*p*r-79*r^2+27*n+69*p-42*r-9)*diff(diff(A[n+1](r,s),r),r)+(4*n^3*p^6-24*n^3* p^5*r+60*n^3*p^4*r^2-80*n^3*p^3*r^3+60*n^3*p^2*r^4-24*n^3*p*r^5+4*n^3*r^6+3*n^2 *p^7-9*n^2*p^6*r-9*n^2*p^5*r^2+75*n^2*p^4*r^3-135*n^2*p^3*r^4+117*n^2*p^2*r^5-\ 51*n^2*p*r^6+9*n^2*r^7+6*n*p^7*r-30*n*p^6*r^2+54*n*p^5*r^3-30*n*p^4*r^4-30*n*p^ 3*r^5+54*n*p^2*r^6-30*n*p*r^7+6*n*r^8+3*p^7*r^2-17*p^6*r^3+39*p^5*r^4-45*p^4*r^ 5+25*p^3*r^6-3*p^2*r^7-3*p*r^8+r^9-46*n^3*p^5+230*n^3*p^4*r-460*n^3*p^3*r^2+460 *n^3*p^2*r^3-230*n^3*p*r^4+46*n^3*r^5-30*n^2*p^6+42*n^2*p^5*r+240*n^2*p^4*r^2-\ 780*n^2*p^3*r^3+930*n^2*p^2*r^4-510*n^2*p*r^5+108*n^2*r^6+6*n*p^7-102*n*p^6*r+ 348*n*p^5*r^2-420*n*p^4*r^3+30*n*p^3*r^4+354*n*p^2*r^5-288*n*p*r^6+72*n*r^7+6*p ^7*r-72*p^6*r^2+260*p^5*r^3-430*p^4*r^4+350*p^3*r^5-116*p^2*r^6-8*p*r^7+10*r^8+ 210*n^3*p^4-840*n^3*p^3*r+1260*n^3*p^2*r^2-840*n^3*p*r^3+210*n^3*r^4+102*n^2*p^ 5+120*n^2*p^4*r-1500*n^2*p^3*r^2+2760*n^2*p^2*r^3-2010*n^2*p*r^4+528*n^2*r^5-72 *n*p^6+636*n*p^5*r-1470*n*p^4*r^2+960*n*p^3*r^3+660*n*p^2*r^4-1068*n*p*r^5+354* n*r^6+3*p^7-93*p^6*r+597*p^5*r^2-1485*p^4*r^3+1725*p^3*r^4-903*p^2*r^5+123*p*r^ 6+33*r^7-488*n^3*p^3+1464*n^3*p^2*r-1464*n^3*p*r^2+488*n^3*r^3-96*n^2*p^4-1080* n^2*p^3*r+3816*n^2*p^2*r^2-4008*n^2*p*r^3+1368*n^2*r^4+342*n*p^5-1902*n*p^4*r+ 2724*n*p^3*r^2-180*n*p^2*r^3-1914*n*p*r^4+930*n*r^5-38*p^6+570*p^5*r-2376*p^4*r ^2+4076*p^3*r^3-3102*p^2*r^4+858*p*r^5+12*r^6+610*n^3*p^2-1220*n^3*p*r+610*n^3* r^2-207*n^2*p^3+2451*n^2*p^2*r-4281*n^2*p*r^2+2037*n^2*r^3-822*n*p^4+2874*n*p^3 *r-1860*n*p^2*r^2-1614*n*p*r^3+1422*n*r^4+194*p^5-1792*p^4*r+5021*p^3*r^2-5641* p^2*r^3+2417*p*r^4-199*r^5-390*n^3*p+390*n^3*r+582*n^2*p^2-2334*n^2*p*r+1752*n^ 2*r^2+1050*n*p^3-1986*n*p^2*r-348*n*p*r^2+1284*n*r^3-516*p^4+3114*p^3*r-5664*p^ 2*r^2+3660*p*r^3-594*r^4+100*n^3-510*n^2*p+810*n^2*r-666*n*p^2+312*n*p*r+654*n* r^2+769*p^3-2973*p^2*r+3129*p*r^2-825*r^3+156*n^2+150*n*p+162*n*r-638*p^2+1426* p*r-632*r^2+12*n+270*p-258*r-44)/(-4+p-r)/(4*n^3*p^5-20*n^3*p^4*r+40*n^3*p^3*r^ 2-40*n^3*p^2*r^3+20*n^3*p*r^4-4*n^3*r^5+9*n^2*p^6-42*n^2*p^5*r+75*n^2*p^4*r^2-\ 60*n^2*p^3*r^3+15*n^2*p^2*r^4+6*n^2*p*r^5-3*n^2*r^6+6*n*p^7-24*n*p^6*r+30*n*p^5 *r^2-30*n*p^3*r^4+24*n*p^2*r^5-6*n*p*r^6+p^8-2*p^7*r-5*p^6*r^2+20*p^5*r^3-25*p^ 4*r^4+14*p^3*r^5-3*p^2*r^6-42*n^3*p^4+168*n^3*p^3*r-252*n^3*p^2*r^2+168*n^3*p*r ^3-42*n^3*r^4-96*n^2*p^5+354*n^2*p^4*r-456*n^2*p^3*r^2+204*n^2*p^2*r^3+24*n^2*p *r^4-30*n^2*r^5-63*n*p^6+186*n*p^5*r-111*n*p^4*r^2-156*n*p^3*r^3+219*n*p^2*r^4-\ 78*n*p*r^5+3*n*r^6-9*p^7+93*p^5*r^2-192*p^4*r^3+153*p^3*r^4-48*p^2*r^5+3*p*r^6+ 172*n^3*p^3-516*n^3*p^2*r+516*n^3*p*r^2-172*n^3*r^3+399*n^2*p^4-1080*n^2*p^3*r+ 846*n^2*p^2*r^2-48*n^2*p*r^3-117*n^2*r^4+252*n*p^5-462*n*p^4*r-156*n*p^3*r^2+ 720*n*p^2*r^3-384*n*p*r^4+30*n*r^5+24*p^6+108*p^5*r-501*p^4*r^2+616*p^3*r^3-282 *p^2*r^4+36*p*r^5-r^6-345*n^3*p^2+690*n^3*p*r-345*n^3*r^2-807*n^2*p^3+1386*n^2* p^2*r-351*n^2*p*r^2-228*n^2*r^3-462*n*p^4+234*n*p^3*r+1035*n*p^2*r^2-924*n*p*r^ 3+117*n*r^4+10*p^5-512*p^4*r+1141*p^3*r^2-796*p^2*r^3+167*p*r^4-10*r^5+342*n^3* p-342*n^3*r+789*n^2*p^2-552*n^2*p*r-237*n^2*r^2+324*n*p^3+606*n*p^2*r-1158*n*p* r^2+228*n*r^3-162*p^4+972*p^3*r-1155*p^2*r^2+384*p*r^3-39*r^4-135*n^3-279*n^2*p -126*n^2*r+84*n*p^2-726*n*p*r+237*n*r^2+304*p^3-828*p^2*r+465*p*r^2-76*r^3-27*n ^2-180*n*p+126*n*r-232*p^2+284*p*r-79*r^2+27*n+69*p-42*r-9)*diff(diff(diff(A[n+ 1](r,s),r),r),r)-(n^3*p^5-5*n^3*p^4*r+10*n^3*p^3*r^2-10*n^3*p^2*r^3+5*n^3*p*r^4 -n^3*r^5+3*n^2*p^5*r-15*n^2*p^4*r^2+30*n^2*p^3*r^3-30*n^2*p^2*r^4+15*n^2*p*r^5-\ 3*n^2*r^6+3*n*p^5*r^2-15*n*p^4*r^3+30*n*p^3*r^4-30*n*p^2*r^5+15*n*p*r^6-3*n*r^7 +p^5*r^3-5*p^4*r^4+10*p^3*r^5-10*p^2*r^6+5*p*r^7-r^8-7*n^3*p^4+28*n^3*p^3*r-42* n^3*p^2*r^2+28*n^3*p*r^3-7*n^3*r^4+3*n^2*p^5-36*n^2*p^4*r+114*n^2*p^3*r^2-156*n ^2*p^2*r^3+99*n^2*p*r^4-24*n^2*r^5+6*n*p^5*r-51*n*p^4*r^2+144*n*p^3*r^3-186*n*p ^2*r^4+114*n*p*r^5-27*n*r^6+3*p^5*r^2-22*p^4*r^3+58*p^3*r^4-72*p^2*r^5+43*p*r^6 -10*r^7+19*n^3*p^3-57*n^3*p^2*r+57*n^3*p*r^2-19*n^3*r^3-21*n^2*p^4+141*n^2*p^3* r-297*n^2*p^2*r^2+255*n^2*p*r^3-78*n^2*r^4+3*n*p^5-57*n*p^4*r+255*n*p^3*r^2-453 *n*p^2*r^3+354*n*p*r^4-102*n*r^5+3*p^5*r-36*p^4*r^2+133*p^3*r^3-213*p^2*r^4+156 *p*r^5-43*r^6-25*n^3*p^2+50*n^3*p*r-25*n^3*r^2+57*n^2*p^3-246*n^2*p^2*r+321*n^2 *p*r^2-132*n^2*r^3-21*n*p^4+198*n*p^3*r-543*n*p^2*r^2+576*n*p*r^3-210*n*r^4+p^5 -26*p^4*r+151*p^3*r^2-332*p^2*r^3+310*p*r^4-104*r^5+16*n^3*p-16*n^3*r-75*n^2*p^ 2+198*n^2*p*r-123*n^2*r^2+57*n*p^3-321*n*p^2*r+519*n*p*r^2-255*n*r^3-7*p^4+85*p ^3*r-288*p^2*r^2+365*p*r^3-155*r^4-4*n^3+48*n^2*p-60*n^2*r-75*n*p^2+246*n*p*r-\ 183*n*r^2+19*p^3-132*p^2*r+255*p*r^2-146*r^3-12*n^2+48*n*p-72*n*r-25*p^2+98*p*r -85*r^2-12*n+16*p-28*r-4)/(4*n^3*p^4-16*n^3*p^3*r+24*n^3*p^2*r^2-16*n^3*p*r^3+4 *n^3*r^4+9*n^2*p^5-33*n^2*p^4*r+42*n^2*p^3*r^2-18*n^2*p^2*r^3-3*n^2*p*r^4+3*n^2 *r^5+6*n*p^6-18*n*p^5*r+12*n*p^4*r^2+12*n*p^3*r^3-18*n*p^2*r^4+6*n*p*r^5+p^7-p^ 6*r-6*p^5*r^2+14*p^4*r^3-11*p^3*r^4+3*p^2*r^5-30*n^3*p^3+90*n^3*p^2*r-90*n^3*p* r^2+30*n^3*r^3-69*n^2*p^4+186*n^2*p^3*r-144*n^2*p^2*r^2+6*n^2*p*r^3+21*n^2*r^4-\ 45*n*p^5+87*n*p^4*r+12*n*p^3*r^2-108*n*p^2*r^3+57*n*p*r^4-3*n*r^5-6*p^6-9*p^5*r +66*p^4*r^2-84*p^3*r^3+36*p^2*r^4-3*p*r^5+82*n^3*p^2-164*n^3*p*r+82*n^3*r^2+192 *n^2*p^3-330*n^2*p^2*r+84*n^2*p*r^2+54*n^2*r^3+117*n*p^4-84*n*p^3*r-204*n*p^2*r ^2+192*n*p*r^3-21*n*r^4+6*p^5+87*p^4*r-216*p^3*r^2+148*p^2*r^3-26*p*r^4+r^5-99* n^3*p+99*n^3*r-231*n^2*p^2+165*n^2*p*r+66*n^2*r^2-111*n*p^3-129*n*p^2*r+294*n*p *r^2-54*n*r^3+28*p^4-223*p^3*r+270*p^2*r^2-82*p*r^3+7*r^4+45*n^3+96*n^2*p+39*n^ 2*r-9*n*p^2+210*n*p*r-66*n*r^2-78*p^3+225*p^2*r-120*p*r^2+18*r^3+9*n^2+57*n*p-\ 39*n*r+70*p^2-83*p*r+22*r^2-9*n-22*p+13*r+3)/(-4+p-r)*diff(diff(diff(diff(A[n+1 ](r,s),r),r),r),r) = 0 (8*n^6+36*n^5*q-24*n^5*s+66*n^4*q^2-84*n^4*q*s+24*n^4*s^2+63*n^3*q^3-114*n^3*q^ 2*s+60*n^3*q*s^2-8*n^3*s^3+33*n^2*q^4-75*n^2*q^3*s+54*n^2*q^2*s^2-12*n^2*q*s^3+ 9*n*q^5-24*n*q^4*s+21*n*q^3*s^2-6*n*q^2*s^3+q^6-3*q^5*s+3*q^4*s^2-q^3*s^3-4*n^5 -8*n^4*q-4*n^4*s-n^3*q^2-28*n^3*q*s+20*n^3*s^2+7*n^2*q^3-45*n^2*q^2*s+48*n^2*q* s^2-12*n^2*s^3+5*n*q^4-26*n*q^3*s+33*n*q^2*s^2-12*n*q*s^3+q^5-5*q^4*s+7*q^3*s^2 -3*q^2*s^3-26*n^4-71*n^3*q+38*n^3*s-69*n^2*q^2+63*n^2*q*s-6*n^2*s^2-28*n*q^3+30 *n*q^2*s+3*n*q*s^2-6*n*s^3-4*q^4+4*q^3*s+3*q^2*s^2-3*q*s^3-7*n^3-27*n^2*q+33*n^ 2*s-24*n*q^2+42*n*q*s-9*n*s^2-6*q^3+12*q^2*s-3*q*s^2-s^3+16*n^2+11*n*q+10*n*s+q ^2+7*q*s-2*s^2+11*n+5*q+s+2)/(n+q-s)^2/(n-s+1)^3*A[n](r,s)-(12*n^7+60*n^6*q-36* n^6*s+123*n^5*q^2-132*n^5*q*s+24*n^5*s^2+132*n^4*q^3-177*n^4*q^2*s+24*n^4*q*s^2 +24*n^4*s^3+78*n^3*q^4-96*n^3*q^3*s-66*n^3*q^2*s^2+120*n^3*q*s^3-36*n^3*s^4+24* n^2*q^5-6*n^2*q^4*s-120*n^2*q^3*s^2+174*n^2*q^2*s^3-84*n^2*q*s^4+12*n^2*s^5+3*n *q^6+12*n*q^5*s-66*n*q^4*s^2+96*n*q^3*s^3-57*n*q^2*s^4+12*n*q*s^5+3*q^6*s-12*q^ 5*s^2+18*q^4*s^3-12*q^3*s^4+3*q^2*s^5-6*n^6-9*n^5*q-18*n^5*s+21*n^4*q^2-129*n^4 *q*s+84*n^4*s^2+60*n^3*q^3-276*n^3*q^2*s+294*n^3*q*s^2-84*n^3*s^3+54*n^2*q^4-\ 252*n^2*q^3*s+342*n^2*q^2*s^2-162*n^2*q*s^3+18*n^2*s^4+21*n*q^5-102*n*q^4*s+156 *n*q^3*s^2-84*n*q^2*s^3+3*n*q*s^4+6*n*s^5+3*q^6-15*q^5*s+24*q^4*s^2-12*q^3*s^3-\ 3*q^2*s^4+3*q*s^5-41*n^5-145*n^4*q+85*n^4*s-194*n^3*q^2+196*n^3*q*s-26*n^3*s^2-\ 120*n^2*q^3+138*n^2*q^2*s+18*n^2*q*s^2-38*n^2*s^3-33*n*q^4+24*n*q^3*s+66*n*q^2* s^2-76*n*q*s^3+19*n*s^4-3*q^5-3*q^4*s+24*q^3*s^2-26*q^2*s^3+7*q*s^4+s^5-4*n^4-\ 40*n^3*q+64*n^3*s-72*n^2*q^2+168*n^2*q*s-72*n^2*s^2-45*n*q^3+126*n*q^2*s-84*n*q *s^2+8*n*s^3-9*q^4+27*q^3*s-18*q^2*s^2-4*q*s^3+4*s^4+28*n^3+54*n^2*q-24*n^2*s+ 27*n*q^2-24*n*s^2+3*q^3+9*q^2*s-18*q*s^2+4*s^3+4*n^2+17*n*q-26*n*s+7*q^2-11*q*s -2*s^2-7*n-q-5*s-2)/(n+q-s)/(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2-n-q+s)/(n-s+1)^3* diff(A[n](r,s),s)+(6*n^6+21*n^5*q-6*n^5*s+27*n^4*q^2-3*n^4*q*s-12*n^4*s^2+15*n^ 3*q^3+18*n^3*q^2*s-42*n^3*q*s^2+12*n^3*s^3+3*n^2*q^4+21*n^2*q^3*s-36*n^2*q^2*s^ 2+6*n^2*q*s^3+6*n^2*s^4+6*n*q^4*s-3*n*q^3*s^2-18*n*q^2*s^3+21*n*q*s^4-6*n*s^5+3 *q^4*s^2-9*q^3*s^3+9*q^2*s^4-3*q*s^5+6*n^5+27*n^4*q-24*n^4*s+42*n^3*q^2-60*n^3* q*s+12*n^3*s^2+27*n^2*q^3-36*n^2*q^2*s-18*n^2*q*s^2+24*n^2*s^3+6*n*q^4+6*n*q^3* s-54*n*q^2*s^2+60*n*q*s^3-18*n*s^4+6*q^4*s-21*q^3*s^2+24*q^2*s^3-9*q*s^4-16*n^4 -25*n^3*q-14*n^3*s-3*n^2*q^2-63*n^2*q*s+42*n^2*s^2+9*n*q^3-60*n*q^2*s+57*n*q*s^ 2-10*n*s^3+3*q^4-15*q^3*s+15*q^2*s^2-q*s^3-2*s^4-22*n^3-45*n^2*q+24*n^2*s-24*n* q^2+6*n*q*s+18*n*s^2-3*q^3-6*q^2*s+15*q*s^2-4*s^3-12*n*q+24*n*s-6*q^2+12*q*s+8* n+2*q+4*s+2)/(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2-n-q+s)/(n-s+1)^3*diff(diff(A[n](r,s ),s),s)-(n^4+n^3*q+2*n^3*s+3*n^2*q*s+3*n*q*s^2-2*n*s^3+q*s^3-s^4+4*n^3+3*n^2*q+ 6*n^2*s+6*n*q*s+3*q*s^2-2*s^3+6*n^2+3*n*q+6*n*s+3*q*s+4*n+q+2*s+1)/(n^3-3*n^2*s +3*n*s^2-s^3+3*n^2-6*n*s+3*s^2+3*n-3*s+1)*diff(diff(diff(A[n](r,s),s),s),s)-(n^ 3+3*n^2*q-3*n^2*s+3*n*q^2-6*n*q*s+3*n*s^2+q^3-3*q^2*s+3*q*s^2-s^3-3*n^2-6*n*q+6 *n*s-3*q^2+6*q*s-3*s^2+2*n+2*q-2*s)*q^3/(n+q-s+1)^3/(n-s+1)^3*A[n+1](r,s)-(3*n^ 6*q^2+14*n^5*q^3-18*n^5*q^2*s+25*n^4*q^4-70*n^4*q^3*s+45*n^4*q^2*s^2+20*n^3*q^5 -100*n^3*q^4*s+140*n^3*q^3*s^2-60*n^3*q^2*s^3+5*n^2*q^6-60*n^2*q^5*s+150*n^2*q^ 4*s^2-140*n^2*q^3*s^3+45*n^2*q^2*s^4-2*n*q^7-10*n*q^6*s+60*n*q^5*s^2-100*n*q^4* s^3+70*n*q^3*s^4-18*n*q^2*s^5-q^8+2*q^7*s+5*q^6*s^2-20*q^5*s^3+25*q^4*s^4-14*q^ 3*s^5+3*q^2*s^6-3*n^6*q-15*n^5*q^2+18*n^5*q*s-24*n^4*q^3+75*n^4*q^2*s-45*n^4*q* s^2-6*n^3*q^4+96*n^3*q^3*s-150*n^3*q^2*s^2+60*n^3*q*s^3+21*n^2*q^5+18*n^2*q^4*s -144*n^2*q^3*s^2+150*n^2*q^2*s^3-45*n^2*q*s^4+21*n*q^6-42*n*q^5*s-18*n*q^4*s^2+ 96*n*q^3*s^3-75*n*q^2*s^4+18*n*q*s^5+6*q^7-21*q^6*s+21*q^5*s^2+6*q^4*s^3-24*q^3 *s^4+15*q^2*s^5-3*q*s^6+n^6+3*n^5*q-6*n^5*s-12*n^4*q^2-15*n^4*q*s+15*n^4*s^2-52 *n^3*q^3+48*n^3*q^2*s+30*n^3*q*s^2-20*n^3*s^3-69*n^2*q^4+156*n^2*q^3*s-72*n^2*q ^2*s^2-30*n^2*q*s^3+15*n^2*s^4-39*n*q^5+138*n*q^4*s-156*n*q^3*s^2+48*n*q^2*s^3+ 15*n*q*s^4-6*n*s^5-8*q^6+39*q^5*s-69*q^4*s^2+52*q^3*s^3-12*q^2*s^4-3*q*s^5+s^6+ n^5+17*n^4*q-5*n^4*s+40*n^3*q^2-68*n^3*q*s+10*n^3*s^2+27*n^2*q^3-120*n^2*q^2*s+ 102*n^2*q*s^2-10*n^2*s^3-3*n*q^4-54*n*q^3*s+120*n*q^2*s^2-68*n*q*s^3+5*n*s^4-6* q^5+3*q^4*s+27*q^3*s^2-40*q^2*s^3+17*q*s^4-s^5-4*n^4+2*n^3*q+16*n^3*s+33*n^2*q^ 2-6*n^2*q*s-24*n^2*s^2+39*n*q^3-66*n*q^2*s+6*n*q*s^2+16*n*s^3+12*q^4-39*q^3*s+ 33*q^2*s^2-2*q*s^3-4*s^4-6*n^3-21*n^2*q+18*n^2*s-9*n*q^2+42*n*q*s-18*n*s^2+4*q^ 3+9*q^2*s-21*q*s^2+6*s^3+n^2-13*n*q-2*n*s-8*q^2+13*q*s+s^2+5*n-q-5*s+2)/(n+q-s+ 1)/(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2+n+q-s)^2/(n-s+1)^3*diff(A[n+1](r,s),s)-(3*n^6 *q+12*n^5*q^2-18*n^5*q*s+15*n^4*q^3-60*n^4*q^2*s+45*n^4*q*s^2-60*n^3*q^3*s+120* n^3*q^2*s^2-60*n^3*q*s^3-15*n^2*q^5+90*n^2*q^3*s^2-120*n^2*q^2*s^3+45*n^2*q*s^4 -12*n*q^6+30*n*q^5*s-60*n*q^3*s^3+60*n*q^2*s^4-18*n*q*s^5-3*q^7+12*q^6*s-15*q^5 *s^2+15*q^3*s^4-12*q^2*s^5+3*q*s^6-3*n^6-12*n^5*q+18*n^5*s-3*n^4*q^2+60*n^4*q*s -45*n^4*s^2+48*n^3*q^3+12*n^3*q^2*s-120*n^3*q*s^2+60*n^3*s^3+87*n^2*q^4-144*n^2 *q^3*s-18*n^2*q^2*s^2+120*n^2*q*s^3-45*n^2*s^4+60*n*q^5-174*n*q^4*s+144*n*q^3*s ^2+12*n*q^2*s^3-60*n*q*s^4+18*n*s^5+15*q^6-60*q^5*s+87*q^4*s^2-48*q^3*s^3-3*q^2 *s^4+12*q*s^5-3*s^6-3*n^5-33*n^4*q+15*n^4*s-99*n^3*q^2+132*n^3*q*s-30*n^3*s^2-\ 129*n^2*q^3+297*n^2*q^2*s-198*n^2*q*s^2+30*n^2*s^3-78*n*q^4+258*n*q^3*s-297*n*q ^2*s^2+132*n*q*s^3-15*n*s^4-18*q^5+78*q^4*s-129*q^3*s^2+99*q^2*s^3-33*q*s^4+3*s ^5+12*n^4+30*n^3*q-48*n^3*s+12*n^2*q^2-90*n^2*q*s+72*n^2*s^2-18*n*q^3-24*n*q^2* s+90*n*q*s^2-48*n*s^3-12*q^4+18*q^3*s+12*q^2*s^2-30*q*s^3+12*s^4+14*n^3+53*n^2* q-42*n^2*s+63*n*q^2-106*n*q*s+42*n*s^2+24*q^3-63*q^2*s+53*q*s^2-14*s^3-3*n^2-2* n*q+6*n*s+3*q^2+2*q*s-3*s^2-3*n-7*q+3*s+2)/(n^2+2*n*q-2*n*s+q^2-2*q*s+s^2+n+q-s )/(n-s+1)^2/(n^3+3*n^2*q-3*n^2*s+3*n*q^2-6*n*q*s+3*n*s^2+q^3-3*q^2*s+3*q*s^2-s^ 3-n-q+s)*diff(diff(A[n+1](r,s),s),s)-(n^4-4*n^3*s-6*n^2*q^2+6*n^2*s^2-8*n*q^3+ 12*n*q^2*s-4*n*s^3-3*q^4+8*q^3*s-6*q^2*s^2+s^4+4*n^3+18*n^2*q-12*n^2*s+24*n*q^2 -36*n*q*s+12*n*s^2+10*q^3-24*q^2*s+18*q*s^2-4*s^3-5*n^2-8*n*q+10*n*s-3*q^2+8*q* s-5*s^2-6*n-8*q+6*s+2)/(n^3+3*n^2*q-3*n^2*s+3*n*q^2-6*n*q*s+3*n*s^2+q^3-3*q^2*s +3*q*s^2-s^3-n-q+s)/(n-s+1)*diff(diff(diff(A[n+1](r,s),s),s),s)+diff(diff(diff( diff(A[n+1](r,s),s),s),s),s) = 0 ------------------------------------------------- This took, 0.948, seconds.