Theorem: For any integers p and q define the Abel-sum type sequence by n ----- k (k - 1 + p) (n - k + q) \ x (r + k) (s - k) A[n](r, s) = ) ---------------------------------------- / k! (n - k)! ----- k = 0 Then we have the following functional recurrence expressing A[n](r.s) in ter\ ms of , A[n - 1](r, s), A[n - 2](r, s) : x (n + r) A[n - 1](r + 1, s - 1) s A[n - 1](r, s) A[n](r, s) = -------------------------------- + ---------------- n n x (s + r) A[n - 2](r + 1, s - 1) - -------------------------------- n and in Maple notation A[n](r,s) = x*(n+r)/n*A[n-1](r+1,s-1)+1/n*s*A[n-1](r,s)-x*(s+r)/n*A[n-2](r+1,s-\ 1) Proof: We claim that: Let a(n,k,r,s) be the summand on the sum defining A[n](r,s) in other words k (k - 1 + p) (n - k + q) x (r + k) (s - k) a(n, k, r, s) = ---------------------------------------- k! (n - k)! and in Maple notation a(n,k,r,s) = 1/k!/(n-k)!*x^k*(r+k)^(k-1+p)*(s-k)^(n-k+q) Then the following identity is true x (s + r) a(n, k, r, s) (n + r + 1) x a(n + 1, k, r, s) ----------------------- - ------------------------------- 2 + n 2 + n (s + 1) a(n + 1, k + 1, r - 1, s + 1) - ------------------------------------- + a(2 + n, k + 1, r - 1, s + 1) = 2 + n 0 and in Maple notation: x*(s+r)/(2+n)*a(n,k,r,s)-(n+r+1)*x/(2+n)*a(n+1,k,r,s)-(s+1)/(2+n)*a(n+1,k+1,r-1 ,s+1)+a(2+n,k+1,r-1,s+1) = 0 The proof of this identity is routine (divide by a(n,k,r,s), simplify each t\ erm,and now each term is a rational function. Now add them all up and \ verify that they add up to zero.) Now sum it from k=0 to k=n, which is the same as from k=-infinity to k=infin\ ity (since it vanishes for k<0 and k>n x (s + r) A[n](r, s) (n + r + 1) x A[n + 1](r, s) -------------------- - ---------------------------- 2 + n 2 + n (s + 1) A[n + 1](r - 1, s + 1) - ------------------------------ + A[2 + n](r - 1, s + 1) = 0 2 + n replacing n by, n - 2, changing variables, and moving a[n](r,s) to the left \ side, yields the statement of the thereom. QED. ------------------------------------------------- This took, 0.125, seconds. Theorem: For any integers p and q define the Abel-sum type sequence by n ----- k (k - 1 + p) (n - k + q) \ x (r + k) (s - k) A[n](r, s) = ) ---------------------------------------- / 2 ----- (k!) (n - k)! k = 0 Then we have the following functional recurrence expressing A[n](r.s) in ter\ ms of , A[n - 1](r, s), A[n - 2](r, s), A[n - 3](r, s) : x (n + r) A[n - 1](r + 1, s - 1) A[n](r, s) = -------------------------------- 2 n 2 (2 n - 2 n s - 2 n + s) s A[n - 1](r, s) + ----------------------------------------- 2 (n - 1 - s) n 2 2 (n + 2 n r - 2 r s - s - n - r) x A[n - 2](r + 1, s - 1) - ---------------------------------------------------------- 2 (n - 1 - s) n 2 (n - s) s A[n - 2](r, s) - ------------------------- 2 (n - 1 - s) n 2 (n r + n s - r s - s ) x A[n - 3](r + 1, s - 1) + ----------------------------------------------- 2 (n - 1 - s) n and in Maple notation A[n](r,s) = x*(n+r)/n^2*A[n-1](r+1,s-1)+(2*n^2-2*n*s-2*n+s)*s/(n-1-s)/n^2*A[n-1 ](r,s)-(n^2+2*n*r-2*r*s-s^2-n-r)*x/(n-1-s)/n^2*A[n-2](r+1,s-1)-(n-s)*s^2/(n-1-s )/n^2*A[n-2](r,s)+(n*r+n*s-r*s-s^2)*x/(n-1-s)/n^2*A[n-3](r+1,s-1) Proof: We claim that: Let a(n,k,r,s) be the summand on the sum defining A[n](r,s) in other words k (k - 1 + p) (n - k + q) x (r + k) (s - k) a(n, k, r, s) = ---------------------------------------- 2 (k!) (n - k)! and in Maple notation a(n,k,r,s) = 1/k!^2/(n-k)!*x^k*(r+k)^(k-1+p)*(s-k)^(n-k+q) Then the following identity is true (s + r) (n - s + 2) a(n, k, r, s) --------------------------------- (n - s + 1) (n + r + 2) 2 2 (n + 2 n r - 2 r s - s + 3 n + 3 r + 2) a(n + 1, k, r, s) - ----------------------------------------------------------- (n - s + 1) (n + r + 2) 2 (s + 1) (n - s + 2) a(n + 1, k + 1, r - 1, s + 1) - -------------------------------------------------- + a(2 + n, k, r, s) x (n - s + 1) (n + r + 2) 2 (s + 1) (2 n - 2 n s + 8 n - 5 s + 7) a(2 + n, k + 1, r - 1, s + 1) + -------------------------------------------------------------------- x (n - s + 1) (n + r + 2) 2 (n + 3) a(n + 3, k + 1, r - 1, s + 1) - -------------------------------------- = 0 x (n + r + 2) and in Maple notation: (s+r)*(n-s+2)/(n-s+1)/(n+r+2)*a(n,k,r,s)-(n^2+2*n*r-2*r*s-s^2+3*n+3*r+2)/(n-s+1 )/(n+r+2)*a(n+1,k,r,s)-(s+1)^2*(n-s+2)/x/(n-s+1)/(n+r+2)*a(n+1,k+1,r-1,s+1)+a(2 +n,k,r,s)+(s+1)*(2*n^2-2*n*s+8*n-5*s+7)/x/(n-s+1)/(n+r+2)*a(2+n,k+1,r-1,s+1)-(n +3)^2/x/(n+r+2)*a(n+3,k+1,r-1,s+1) = 0 The proof of this identity is routine (divide by a(n,k,r,s), simplify each t\ erm,and now each term is a rational function. Now add them all up and \ verify that they add up to zero.) Now sum it from k=0 to k=n, which is the same as from k=-infinity to k=infin\ ity (since it vanishes for k<0 and k>n (s + r) (n - s + 2) A[n](r, s) ------------------------------ (n - s + 1) (n + r + 2) 2 2 (n + 2 n r - 2 r s - s + 3 n + 3 r + 2) A[n + 1](r, s) - -------------------------------------------------------- (n - s + 1) (n + r + 2) 2 (s + 1) (n - s + 2) A[n + 1](r - 1, s + 1) - ------------------------------------------- + A[2 + n](r, s) x (n - s + 1) (n + r + 2) 2 (s + 1) (2 n - 2 n s + 8 n - 5 s + 7) A[2 + n](r - 1, s + 1) + ------------------------------------------------------------- x (n - s + 1) (n + r + 2) 2 (n + 3) A[n + 3](r - 1, s + 1) - ------------------------------- = 0 x (n + r + 2) replacing n by, n - 3, changing variables, and moving a[n](r,s) to the left \ side, yields the statement of the thereom. QED. ------------------------------------------------- This took, 0.117, seconds. Theorem: For any integers p and q define the Abel-sum type sequence by n ----- k (k - 1 + p) (n - k + q) \ x (r + k) (s - k) A[n](r, s) = ) ---------------------------------------- / 3 ----- (k!) (n - k)! k = 0 Then we have the following functional recurrence expressing A[n](r.s) in ter\ ms of , A[n - 1](r, s), A[n - 2](r, s), A[n - 3](r, s), A[n - 4](r, s) : x (n + r) A[n - 1](r + 1, s - 1) A[n](r, s) = -------------------------------- + 3 n 4 3 2 2 3 2 2 2 2 (3 n - 6 n s + 3 n s - 6 n + 9 n s - 3 n s + 3 n - 3 n s + s ) s / 2 3 3 2 2 A[n - 1](r, s) / ((n - 1 - s) n ) - (2 n + 3 n r - 3 n s - 6 n r s / 2 3 2 + 3 r s + s - 3 n - 3 n r + 3 n s + 3 r s + n + r) x / 2 3 4 3 2 2 A[n - 2](r + 1, s - 1) / ((n - 1 - s) n ) - (3 n - 9 n s + 9 n s / 3 3 2 2 3 2 2 2 - 3 n s - 9 n + 21 n s - 15 n s + 3 s + 6 n - 10 n s + 4 s ) s / 2 3 4 3 3 A[n - 2](r, s) / ((n - 1 - s) (n - 2 - s) n ) + (n + 3 n r - n s / 2 2 2 2 3 3 4 3 2 - 9 n r s - 3 n s + 9 n r s + 5 n s - 3 r s - 2 s - 3 n - 6 n r 2 2 2 3 2 + 3 n s + 12 n r s + 3 n s - 6 r s - 3 s + 2 n + 2 n r - 2 n s / 2 3 - 2 r s) x A[n - 3](r + 1, s - 1) / ((n - 1 - s) (n - 2 - s) n ) / 2 2 3 (n - 2 n s + s ) s A[n - 3](r, s) + ----------------------------------- 3 (n - 2 - s) (n - 1 - s) n 2 2 2 2 3 (n r + n s - 2 n r s - 2 n s + r s + s ) x A[n - 4](r + 1, s - 1) - --------------------------------------------------------------------- 3 (n - 2 - s) (n - 1 - s) n and in Maple notation A[n](r,s) = x*(n+r)/n^3*A[n-1](r+1,s-1)+(3*n^4-6*n^3*s+3*n^2*s^2-6*n^3+9*n^2*s-\ 3*n*s^2+3*n^2-3*n*s+s^2)*s/(n-1-s)^2/n^3*A[n-1](r,s)-(2*n^3+3*n^2*r-3*n^2*s-6*n *r*s+3*r*s^2+s^3-3*n^2-3*n*r+3*n*s+3*r*s+n+r)*x/(n-1-s)^2/n^3*A[n-2](r+1,s-1)-( 3*n^4-9*n^3*s+9*n^2*s^2-3*n*s^3-9*n^3+21*n^2*s-15*n*s^2+3*s^3+6*n^2-10*n*s+4*s^ 2)*s^2/(n-1-s)^2/(n-2-s)/n^3*A[n-2](r,s)+(n^4+3*n^3*r-n^3*s-9*n^2*r*s-3*n^2*s^2 +9*n*r*s^2+5*n*s^3-3*r*s^3-2*s^4-3*n^3-6*n^2*r+3*n^2*s+12*n*r*s+3*n*s^2-6*r*s^2 -3*s^3+2*n^2+2*n*r-2*n*s-2*r*s)*x/(n-1-s)^2/(n-2-s)/n^3*A[n-3](r+1,s-1)+(n^2-2* n*s+s^2)*s^3/(n-2-s)/(n-1-s)/n^3*A[n-3](r,s)-(n^2*r+n^2*s-2*n*r*s-2*n*s^2+r*s^2 +s^3)*x/(n-2-s)/(n-1-s)/n^3*A[n-4](r+1,s-1) Proof: We claim that: Let a(n,k,r,s) be the summand on the sum defining A[n](r,s) in other words k (k - 1 + p) (n - k + q) x (r + k) (s - k) a(n, k, r, s) = ---------------------------------------- 3 (k!) (n - k)! and in Maple notation a(n,k,r,s) = 1/k!^3/(n-k)!*x^k*(r+k)^(k-1+p)*(s-k)^(n-k+q) Then the following identity is true 2 (n - s + 3) (s + r) a(n, k, r, s) 3 2 - ----------------------------------- + (n - s + 3) (n + 3 n r - 6 n r s (n + r + 3) (n - s + 2) (n - s + 1) 2 2 3 2 2 - 3 n s + 3 r s + 2 s + 6 n + 12 n r - 12 r s - 6 s + 11 n + 11 r + 6 / 2 ) a(n + 1, k, r, s) / ((n - s + 2) (n + r + 3) (n - s + 1)) / 3 2 (s + 1) (n - s + 3) a(n + 1, k + 1, r - 1, s + 1) 3 2 + --------------------------------------------------- - (2 n + 3 n r x (n + r + 3) (n - s + 2) (n - s + 1) 2 2 3 2 - 3 n s - 6 n r s + 3 r s + s + 15 n + 15 n r - 15 n s - 15 r s + 37 n / 2 + 19 r - 18 s + 30) a(2 + n, k, r, s) / ((n - s + 2) (n + r + 3)) - / 2 (s + 1) (n - s + 3) 3 2 2 2 2 (3 n - 6 n s + 3 n s + 21 n - 30 n s + 9 s + 45 n - 34 s + 29) / 2 a(2 + n, k + 1, r - 1, s + 1) / (x (n - s + 1) (n - s + 2) (n + r + 3)) / 4 3 2 2 3 2 + a(n + 3, k, r, s) + (s + 1) (3 n - 6 n s + 3 n s + 36 n - 57 n s 2 2 2 + 21 n s + 159 n - 177 n s + 37 s + 306 n - 178 s + 217) / 2 a(n + 3, k + 1, r - 1, s + 1) / (x (n - s + 2) (n + r + 3)) / 3 (4 + n) a(4 + n, k + 1, r - 1, s + 1) - -------------------------------------- = 0 x (n + r + 3) and in Maple notation: -(n-s+3)^2*(s+r)/(n+r+3)/(n-s+2)/(n-s+1)*a(n,k,r,s)+(n-s+3)*(n^3+3*n^2*r-6*n*r* s-3*n*s^2+3*r*s^2+2*s^3+6*n^2+12*n*r-12*r*s-6*s^2+11*n+11*r+6)/(n-s+2)^2/(n+r+3 )/(n-s+1)*a(n+1,k,r,s)+(s+1)^3*(n-s+3)^2/x/(n+r+3)/(n-s+2)/(n-s+1)*a(n+1,k+1,r-\ 1,s+1)-(2*n^3+3*n^2*r-3*n^2*s-6*n*r*s+3*r*s^2+s^3+15*n^2+15*n*r-15*n*s-15*r*s+ 37*n+19*r-18*s+30)/(n-s+2)^2/(n+r+3)*a(2+n,k,r,s)-(s+1)^2*(n-s+3)*(3*n^3-6*n^2* s+3*n*s^2+21*n^2-30*n*s+9*s^2+45*n-34*s+29)/x/(n-s+1)/(n-s+2)^2/(n+r+3)*a(2+n,k +1,r-1,s+1)+a(n+3,k,r,s)+(s+1)*(3*n^4-6*n^3*s+3*n^2*s^2+36*n^3-57*n^2*s+21*n*s^ 2+159*n^2-177*n*s+37*s^2+306*n-178*s+217)/x/(n-s+2)^2/(n+r+3)*a(n+3,k+1,r-1,s+1 )-(4+n)^3/x/(n+r+3)*a(4+n,k+1,r-1,s+1) = 0 The proof of this identity is routine (divide by a(n,k,r,s), simplify each t\ erm,and now each term is a rational function. Now add them all up and \ verify that they add up to zero.) Now sum it from k=0 to k=n, which is the same as from k=-infinity to k=infin\ ity (since it vanishes for k<0 and k>n 2 (n - s + 3) (s + r) A[n](r, s) 3 2 - ----------------------------------- + (n - s + 3) (n + 3 n r - 6 n r s (n + r + 3) (n - s + 2) (n - s + 1) 2 2 3 2 2 - 3 n s + 3 r s + 2 s + 6 n + 12 n r - 12 r s - 6 s + 11 n + 11 r + 6 / 2 ) A[n + 1](r, s) / ((n - s + 2) (n + r + 3) (n - s + 1)) / 3 2 (s + 1) (n - s + 3) A[n + 1](r - 1, s + 1) 3 2 2 + -------------------------------------------- - (2 n + 3 n r - 3 n s x (n + r + 3) (n - s + 2) (n - s + 1) 2 3 2 - 6 n r s + 3 r s + s + 15 n + 15 n r - 15 n s - 15 r s + 37 n + 19 r / 2 2 - 18 s + 30) A[2 + n](r, s) / ((n - s + 2) (n + r + 3)) - (s + 1) / (n - s + 3) 3 2 2 2 2 (3 n - 6 n s + 3 n s + 21 n - 30 n s + 9 s + 45 n - 34 s + 29) / 2 A[2 + n](r - 1, s + 1) / (x (n - s + 1) (n - s + 2) (n + r + 3)) / 4 3 2 2 3 2 + A[n + 3](r, s) + (s + 1) (3 n - 6 n s + 3 n s + 36 n - 57 n s 2 2 2 + 21 n s + 159 n - 177 n s + 37 s + 306 n - 178 s + 217) / 2 A[n + 3](r - 1, s + 1) / (x (n - s + 2) (n + r + 3)) / 3 (4 + n) A[4 + n](r - 1, s + 1) - ------------------------------- = 0 x (n + r + 3) replacing n by, n - 4, changing variables, and moving a[n](r,s) to the left \ side, yields the statement of the thereom. QED. ------------------------------------------------- This took, 0.533, seconds. Theorem: For any integers p and q define the Abel-sum type sequence by n ----- k (k - 1 + p) (n - k + q) \ x (r + k) (s - k) A[n](r, s) = ) ---------------------------------------- / 4 ----- (k!) (n - k)! k = 0 Then we have the following functional recurrence expressing A[n](r.s) in ter\ ms of , A[n - 1](r, s), A[n - 2](r, s), A[n - 3](r, s), A[n - 4](r, s), A[n - 5](r, s) : x (n + r) A[n - 1](r + 1, s - 1) 6 5 4 2 A[n](r, s) = -------------------------------- + s (4 n - 12 n s + 12 n s 4 n 3 3 5 4 3 2 2 3 4 3 - 4 n s - 12 n + 30 n s - 24 n s + 6 n s + 12 n - 24 n s 2 2 3 3 2 2 3 / 4 + 16 n s - 4 n s - 4 n + 6 n s - 4 n s + s ) A[n - 1](r, s) / (n / 3 4 3 3 2 2 2 2 (n - 1 - s) ) - x (3 n + 4 n r - 8 n s - 12 n r s + 6 n s + 12 n r s 3 4 3 2 2 2 2 - 4 r s - s - 6 n - 6 n r + 12 n s + 12 n r s - 6 n s - 6 r s 2 / 4 + 4 n + 4 n r - 4 n s - 4 r s - n - r) A[n - 2](r + 1, s - 1) / (n / 3 2 7 6 5 2 4 3 3 4 (n - 1 - s) ) - s (6 n - 30 n s + 60 n s - 60 n s + 30 n s 2 5 6 5 4 2 3 3 2 4 5 - 6 n s - 36 n + 156 n s - 264 n s + 216 n s - 84 n s + 12 n s 5 4 3 2 2 3 4 5 4 + 78 n - 286 n s + 397 n s - 255 n s + 73 n s - 7 s - 72 n 3 2 2 3 4 3 2 2 + 216 n s - 234 n s + 108 n s - 18 s + 24 n - 56 n s + 44 n s 3 / 4 3 2 6 - 12 s ) A[n - 2](r, s) / (n (n - 1 - s) (n - 2 - s) ) + x (3 n / 5 5 4 4 2 3 2 2 3 + 6 n r - 12 n s - 30 n r s + 15 n s + 60 n r s - 60 n r s 2 4 4 5 5 6 5 4 - 15 n s + 30 n r s + 12 n s - 6 r s - 3 s - 15 n - 24 n r 4 3 3 2 2 2 2 3 3 + 51 n s + 96 n r s - 54 n s - 144 n r s + 6 n s + 96 n r s 4 4 5 4 3 3 2 + 21 n s - 24 r s - 9 s + 26 n + 33 n r - 71 n s - 99 n r s 2 2 2 3 3 4 3 2 + 57 n s + 99 n r s - 5 n s - 33 r s - 7 s - 18 n - 18 n r 2 2 2 2 + 36 n s + 36 n r s - 18 n s - 18 r s + 4 n + 4 n r - 4 n s - 4 r s) / 4 3 2 3 6 A[n - 3](r + 1, s - 1) / (n (n - 1 - s) (n - 2 - s) ) + 2 s (2 n / 5 4 2 3 3 2 4 5 5 4 - 10 n s + 20 n s - 20 n s + 10 n s - 2 n s - 12 n + 51 n s 3 2 2 3 4 5 4 3 2 2 - 84 n s + 66 n s - 24 n s + 3 s + 22 n - 77 n s + 99 n s 3 4 3 2 2 3 / - 55 n s + 11 s - 12 n + 33 n s - 30 n s + 9 s ) A[n - 3](r, s) / ( / 4 2 2 6 5 5 n (n - 1 - s) (n - 2 - s) (n - 3 - s)) - x (n + 4 n r - 2 n s 4 4 2 3 2 3 3 2 3 2 4 - 20 n r s - 5 n s + 40 n r s + 20 n s - 40 n r s - 25 n s 4 5 5 6 5 4 4 + 20 n r s + 14 n s - 4 r s - 3 s - 6 n - 18 n r + 12 n s 3 3 2 2 2 2 3 3 4 + 72 n r s + 12 n s - 108 n r s - 48 n s + 72 n r s + 42 n s 4 5 4 3 3 2 2 - 18 r s - 12 s + 11 n + 22 n r - 22 n s - 66 n r s + 66 n r s 3 3 4 3 2 2 2 + 22 n s - 22 r s - 11 s - 6 n - 6 n r + 12 n s + 12 n r s - 6 n s 2 / 4 2 2 - 6 r s ) A[n - 4](r + 1, s - 1) / (n (n - 1 - s) (n - 2 - s) / 4 3 2 2 3 s (n - 3 n s + 3 n s - s ) A[n - 4](r, s) (n - 3 - s)) - --------------------------------------------- + x 4 n (n - 1 - s) (n - 2 - s) (n - 3 - s) 3 3 2 2 2 2 3 3 4 (n r + n s - 3 n r s - 3 n s + 3 n r s + 3 n s - r s - s ) / 4 A[n - 5](r + 1, s - 1) / (n (n - 1 - s) (n - 2 - s) (n - 3 - s)) / and in Maple notation A[n](r,s) = x*(n+r)/n^4*A[n-1](r+1,s-1)+s*(4*n^6-12*n^5*s+12*n^4*s^2-4*n^3*s^3-\ 12*n^5+30*n^4*s-24*n^3*s^2+6*n^2*s^3+12*n^4-24*n^3*s+16*n^2*s^2-4*n*s^3-4*n^3+6 *n^2*s-4*n*s^2+s^3)/n^4/(n-1-s)^3*A[n-1](r,s)-x*(3*n^4+4*n^3*r-8*n^3*s-12*n^2*r *s+6*n^2*s^2+12*n*r*s^2-4*r*s^3-s^4-6*n^3-6*n^2*r+12*n^2*s+12*n*r*s-6*n*s^2-6*r *s^2+4*n^2+4*n*r-4*n*s-4*r*s-n-r)/n^4/(n-1-s)^3*A[n-2](r+1,s-1)-s^2*(6*n^7-30*n ^6*s+60*n^5*s^2-60*n^4*s^3+30*n^3*s^4-6*n^2*s^5-36*n^6+156*n^5*s-264*n^4*s^2+ 216*n^3*s^3-84*n^2*s^4+12*n*s^5+78*n^5-286*n^4*s+397*n^3*s^2-255*n^2*s^3+73*n*s ^4-7*s^5-72*n^4+216*n^3*s-234*n^2*s^2+108*n*s^3-18*s^4+24*n^3-56*n^2*s+44*n*s^2 -12*s^3)/n^4/(n-1-s)^3/(n-2-s)^2*A[n-2](r,s)+x*(3*n^6+6*n^5*r-12*n^5*s-30*n^4*r *s+15*n^4*s^2+60*n^3*r*s^2-60*n^2*r*s^3-15*n^2*s^4+30*n*r*s^4+12*n*s^5-6*r*s^5-\ 3*s^6-15*n^5-24*n^4*r+51*n^4*s+96*n^3*r*s-54*n^3*s^2-144*n^2*r*s^2+6*n^2*s^3+96 *n*r*s^3+21*n*s^4-24*r*s^4-9*s^5+26*n^4+33*n^3*r-71*n^3*s-99*n^2*r*s+57*n^2*s^2 +99*n*r*s^2-5*n*s^3-33*r*s^3-7*s^4-18*n^3-18*n^2*r+36*n^2*s+36*n*r*s-18*n*s^2-\ 18*r*s^2+4*n^2+4*n*r-4*n*s-4*r*s)/n^4/(n-1-s)^3/(n-2-s)^2*A[n-3](r+1,s-1)+2*s^3 *(2*n^6-10*n^5*s+20*n^4*s^2-20*n^3*s^3+10*n^2*s^4-2*n*s^5-12*n^5+51*n^4*s-84*n^ 3*s^2+66*n^2*s^3-24*n*s^4+3*s^5+22*n^4-77*n^3*s+99*n^2*s^2-55*n*s^3+11*s^4-12*n ^3+33*n^2*s-30*n*s^2+9*s^3)/n^4/(n-1-s)^2/(n-2-s)^2/(n-3-s)*A[n-3](r,s)-x*(n^6+ 4*n^5*r-2*n^5*s-20*n^4*r*s-5*n^4*s^2+40*n^3*r*s^2+20*n^3*s^3-40*n^2*r*s^3-25*n^ 2*s^4+20*n*r*s^4+14*n*s^5-4*r*s^5-3*s^6-6*n^5-18*n^4*r+12*n^4*s+72*n^3*r*s+12*n ^3*s^2-108*n^2*r*s^2-48*n^2*s^3+72*n*r*s^3+42*n*s^4-18*r*s^4-12*s^5+11*n^4+22*n ^3*r-22*n^3*s-66*n^2*r*s+66*n*r*s^2+22*n*s^3-22*r*s^3-11*s^4-6*n^3-6*n^2*r+12*n ^2*s+12*n*r*s-6*n*s^2-6*r*s^2)/n^4/(n-1-s)^2/(n-2-s)^2/(n-3-s)*A[n-4](r+1,s-1)- s^4*(n^3-3*n^2*s+3*n*s^2-s^3)/n^4/(n-1-s)/(n-2-s)/(n-3-s)*A[n-4](r,s)+x*(n^3*r+ n^3*s-3*n^2*r*s-3*n^2*s^2+3*n*r*s^2+3*n*s^3-r*s^3-s^4)/n^4/(n-1-s)/(n-2-s)/(n-3 -s)*A[n-5](r+1,s-1) Proof: We claim that: Let a(n,k,r,s) be the summand on the sum defining A[n](r,s) in other words k (k - 1 + p) (n - k + q) x (r + k) (s - k) a(n, k, r, s) = ---------------------------------------- 4 (k!) (n - k)! and in Maple notation a(n,k,r,s) = 1/k!^4/(n-k)!*x^k*(r+k)^(k-1+p)*(s-k)^(n-k+q) Then the following identity is true 3 (n - s + 4) (s + r) a(n, k, r, s) 2 4 3 ----------------------------------------------- - (n - s + 4) (n + 4 n r (n - s + 3) (n - s + 2) (n + r + 4) (n - s + 1) 2 2 2 2 3 3 4 3 - 12 n r s - 6 n s + 12 n r s + 8 n s - 4 r s - 3 s + 10 n 2 2 2 3 2 + 30 n r - 60 n r s - 30 n s + 30 r s + 20 s + 35 n + 70 n r - 70 r s 2 / - 35 s + 50 n + 50 r + 24) a(n + 1, k, r, s) / ((n - s + 1) / 2 2 (n - s + 2) (n - s + 3) (n + r + 4)) 4 3 (s + 1) (n - s + 4) a(n + 1, k + 1, r - 1, s + 1) 5 - --------------------------------------------------- + (n - s + 4) (3 n x (n - s + 3) (n - s + 2) (n + r + 4) (n - s + 1) 4 4 3 3 2 2 2 2 3 3 + 6 n r - 9 n s - 24 n r s + 6 n s + 36 n r s + 6 n s - 24 n r s 4 4 5 4 3 3 2 - 9 n s + 6 r s + 3 s + 45 n + 72 n r - 108 n s - 216 n r s 2 2 2 3 3 4 3 2 + 54 n s + 216 n r s + 36 n s - 72 r s - 27 s + 266 n + 321 n r 2 2 2 3 2 - 477 n s - 642 n r s + 156 n s + 321 r s + 55 s + 774 n + 630 n r 2 - 918 n s - 630 r s + 144 s + 1108 n + 460 r - 648 s + 624) / 3 2 3 a(2 + n, k, r, s) / ((n - s + 3) (n + r + 4) (n - s + 2) ) + 2 (s + 1) / 4 3 2 2 3 3 2 2 3 (2 n - 6 n s + 6 n s - 2 n s + 22 n - 51 n s + 36 n s - 7 s 2 2 2 + 85 n - 135 n s + 50 s + 135 n - 110 s + 73) (n - s + 4) / 2 a(2 + n, k + 1, r - 1, s + 1) / (x (n - s + 1) (n + r + 4) (n - s + 3) / 2 4 3 3 2 2 2 2 (n - s + 2) ) - (3 n + 4 n r - 8 n s - 12 n r s + 6 n s + 12 n r s 3 4 3 2 2 2 2 - 4 r s - s + 42 n + 42 n r - 84 n s - 84 n r s + 42 n s + 42 r s 2 2 + 220 n + 148 n r - 292 n s - 148 r s + 72 s + 511 n + 175 r - 336 s / 3 2 + 444) a(n + 3, k, r, s) / ((n - s + 3) (n + r + 4)) - (s + 1) / 6 5 4 2 3 3 2 4 5 (n - s + 4) (6 n - 24 n s + 36 n s - 24 n s + 6 n s + 120 n 4 3 2 2 3 4 4 3 - 408 n s + 504 n s - 264 n s + 48 n s + 984 n - 2728 n s 2 2 3 4 3 2 2 + 2601 n s - 954 n s + 97 s + 4232 n - 8958 n s + 5850 n s 3 2 2 - 1124 s + 10065 n - 14438 n s + 4833 s + 12550 n - 9134 s + 6412) / 2 3 a(n + 3, k + 1, r - 1, s + 1) / (x (n - s + 2) (n - s + 3) (n + r + 4)) / 2 4 + a(4 + n, k, r, s) + (s + 1) (2 n - 2 n s + 16 n - 9 s + 31) (2 n 3 2 2 3 2 2 2 2 - 4 n s + 2 n s + 32 n - 50 n s + 18 n s + 190 n - 206 n s + 41 s / 3 + 496 n - 278 s + 481) a(4 + n, k + 1, r - 1, s + 1) / (x (n - s + 3) / 4 (n + 5) a(n + 5, k + 1, r - 1, s + 1) (n + r + 4)) - -------------------------------------- = 0 x (n + r + 4) and in Maple notation: (n-s+4)^3*(s+r)/(n-s+3)/(n-s+2)/(n+r+4)/(n-s+1)*a(n,k,r,s)-(n-s+4)^2*(n^4+4*n^3 *r-12*n^2*r*s-6*n^2*s^2+12*n*r*s^2+8*n*s^3-4*r*s^3-3*s^4+10*n^3+30*n^2*r-60*n*r *s-30*n*s^2+30*r*s^2+20*s^3+35*n^2+70*n*r-70*r*s-35*s^2+50*n+50*r+24)/(n-s+1)/( n-s+2)^2/(n-s+3)^2/(n+r+4)*a(n+1,k,r,s)-(s+1)^4*(n-s+4)^3/x/(n-s+3)/(n-s+2)/(n+ r+4)/(n-s+1)*a(n+1,k+1,r-1,s+1)+(n-s+4)*(3*n^5+6*n^4*r-9*n^4*s-24*n^3*r*s+6*n^3 *s^2+36*n^2*r*s^2+6*n^2*s^3-24*n*r*s^3-9*n*s^4+6*r*s^4+3*s^5+45*n^4+72*n^3*r-\ 108*n^3*s-216*n^2*r*s+54*n^2*s^2+216*n*r*s^2+36*n*s^3-72*r*s^3-27*s^4+266*n^3+ 321*n^2*r-477*n^2*s-642*n*r*s+156*n*s^2+321*r*s^2+55*s^3+774*n^2+630*n*r-918*n* s-630*r*s+144*s^2+1108*n+460*r-648*s+624)/(n-s+3)^3/(n+r+4)/(n-s+2)^2*a(2+n,k,r ,s)+2*(s+1)^3*(2*n^4-6*n^3*s+6*n^2*s^2-2*n*s^3+22*n^3-51*n^2*s+36*n*s^2-7*s^3+ 85*n^2-135*n*s+50*s^2+135*n-110*s+73)*(n-s+4)^2/x/(n-s+1)/(n+r+4)/(n-s+3)^2/(n- s+2)^2*a(2+n,k+1,r-1,s+1)-(3*n^4+4*n^3*r-8*n^3*s-12*n^2*r*s+6*n^2*s^2+12*n*r*s^ 2-4*r*s^3-s^4+42*n^3+42*n^2*r-84*n^2*s-84*n*r*s+42*n*s^2+42*r*s^2+220*n^2+148*n *r-292*n*s-148*r*s+72*s^2+511*n+175*r-336*s+444)/(n-s+3)^3/(n+r+4)*a(n+3,k,r,s) -(s+1)^2*(n-s+4)*(6*n^6-24*n^5*s+36*n^4*s^2-24*n^3*s^3+6*n^2*s^4+120*n^5-408*n^ 4*s+504*n^3*s^2-264*n^2*s^3+48*n*s^4+984*n^4-2728*n^3*s+2601*n^2*s^2-954*n*s^3+ 97*s^4+4232*n^3-8958*n^2*s+5850*n*s^2-1124*s^3+10065*n^2-14438*n*s+4833*s^2+ 12550*n-9134*s+6412)/x/(n-s+2)^2/(n-s+3)^3/(n+r+4)*a(n+3,k+1,r-1,s+1)+a(4+n,k,r ,s)+(s+1)*(2*n^2-2*n*s+16*n-9*s+31)*(2*n^4-4*n^3*s+2*n^2*s^2+32*n^3-50*n^2*s+18 *n*s^2+190*n^2-206*n*s+41*s^2+496*n-278*s+481)/x/(n-s+3)^3/(n+r+4)*a(4+n,k+1,r-\ 1,s+1)-(n+5)^4/x/(n+r+4)*a(n+5,k+1,r-1,s+1) = 0 The proof of this identity is routine (divide by a(n,k,r,s), simplify each t\ erm,and now each term is a rational function. Now add them all up and \ verify that they add up to zero.) Now sum it from k=0 to k=n, which is the same as from k=-infinity to k=infin\ ity (since it vanishes for k<0 and k>n 3 (n - s + 4) (s + r) A[n](r, s) 2 4 3 ----------------------------------------------- - (n - s + 4) (n + 4 n r (n - s + 3) (n - s + 2) (n + r + 4) (n - s + 1) 2 2 2 2 3 3 4 3 - 12 n r s - 6 n s + 12 n r s + 8 n s - 4 r s - 3 s + 10 n 2 2 2 3 2 + 30 n r - 60 n r s - 30 n s + 30 r s + 20 s + 35 n + 70 n r - 70 r s 2 / 2 - 35 s + 50 n + 50 r + 24) A[n + 1](r, s) / ((n - s + 1) (n - s + 2) / 2 (n - s + 3) (n + r + 4)) 4 3 (s + 1) (n - s + 4) A[n + 1](r - 1, s + 1) 5 - ------------------------------------------------- + (n - s + 4) (3 n x (n - s + 3) (n - s + 2) (n + r + 4) (n - s + 1) 4 4 3 3 2 2 2 2 3 3 + 6 n r - 9 n s - 24 n r s + 6 n s + 36 n r s + 6 n s - 24 n r s 4 4 5 4 3 3 2 - 9 n s + 6 r s + 3 s + 45 n + 72 n r - 108 n s - 216 n r s 2 2 2 3 3 4 3 2 + 54 n s + 216 n r s + 36 n s - 72 r s - 27 s + 266 n + 321 n r 2 2 2 3 2 - 477 n s - 642 n r s + 156 n s + 321 r s + 55 s + 774 n + 630 n r 2 - 918 n s - 630 r s + 144 s + 1108 n + 460 r - 648 s + 624) / 3 2 3 A[2 + n](r, s) / ((n - s + 3) (n + r + 4) (n - s + 2) ) + 2 (s + 1) ( / 4 3 2 2 3 3 2 2 3 2 2 n - 6 n s + 6 n s - 2 n s + 22 n - 51 n s + 36 n s - 7 s + 85 n 2 2 - 135 n s + 50 s + 135 n - 110 s + 73) (n - s + 4) / 2 A[2 + n](r - 1, s + 1) / (x (n - s + 1) (n + r + 4) (n - s + 3) / 2 4 3 3 2 2 2 2 (n - s + 2) ) - (3 n + 4 n r - 8 n s - 12 n r s + 6 n s + 12 n r s 3 4 3 2 2 2 2 - 4 r s - s + 42 n + 42 n r - 84 n s - 84 n r s + 42 n s + 42 r s 2 2 + 220 n + 148 n r - 292 n s - 148 r s + 72 s + 511 n + 175 r - 336 s / 3 2 + 444) A[n + 3](r, s) / ((n - s + 3) (n + r + 4)) - (s + 1) / 6 5 4 2 3 3 2 4 5 (n - s + 4) (6 n - 24 n s + 36 n s - 24 n s + 6 n s + 120 n 4 3 2 2 3 4 4 3 - 408 n s + 504 n s - 264 n s + 48 n s + 984 n - 2728 n s 2 2 3 4 3 2 2 + 2601 n s - 954 n s + 97 s + 4232 n - 8958 n s + 5850 n s 3 2 2 - 1124 s + 10065 n - 14438 n s + 4833 s + 12550 n - 9134 s + 6412) / 2 3 A[n + 3](r - 1, s + 1) / (x (n - s + 2) (n - s + 3) (n + r + 4)) / 2 4 3 + A[4 + n](r, s) + (s + 1) (2 n - 2 n s + 16 n - 9 s + 31) (2 n - 4 n s 2 2 3 2 2 2 2 + 2 n s + 32 n - 50 n s + 18 n s + 190 n - 206 n s + 41 s + 496 n / 3 - 278 s + 481) A[4 + n](r - 1, s + 1) / (x (n - s + 3) (n + r + 4)) / 4 (n + 5) A[n + 5](r - 1, s + 1) - ------------------------------- = 0 x (n + r + 4) replacing n by, n - 5, changing variables, and moving a[n](r,s) to the left \ side, yields the statement of the thereom. QED. ------------------------------------------------- This took, 2.188, seconds. Theorem: For any integers p and q define the Abel-sum type sequence by n ----- k (k - 1 + p) (n - k + q) \ x (r + k) (s - k) A[n](r, s) = ) ---------------------------------------- / 5 ----- (k!) (n - k)! k = 0 Then we have the following functional recurrence expressing A[n](r.s) in ter\ ms of , A[n - 1](r, s), A[n - 2](r, s), A[n - 3](r, s), A[n - 4](r, s), A[n - 5](r, s), A[n - 6](r, s) : x (n + r) A[n - 1](r + 1, -1 + s) 8 7 6 2 A[n](r, s) = --------------------------------- + s (5 n - 20 n s + 30 n s 5 n 5 3 4 4 7 6 5 2 4 3 3 4 - 20 n s + 5 n s - 20 n + 70 n s - 90 n s + 50 n s - 10 n s 6 5 4 2 3 3 2 4 5 4 + 30 n - 90 n s + 100 n s - 50 n s + 10 n s - 20 n + 50 n s 3 2 2 3 4 4 3 2 2 3 4 - 50 n s + 25 n s - 5 n s + 5 n - 10 n s + 10 n s - 5 n s + s ) / 5 4 5 4 4 A[n - 1](r, s) / (n (n - 1 - s) ) - x (4 n + 5 n r - 15 n s / 3 3 2 2 2 2 3 3 4 5 - 20 n r s + 20 n s + 30 n r s - 10 n s - 20 n r s + 5 r s + s 4 3 3 2 2 2 2 3 - 10 n - 10 n r + 30 n s + 30 n r s - 30 n s - 30 n r s + 10 n s 3 3 2 2 2 2 + 10 r s + 10 n + 10 n r - 20 n s - 20 n r s + 10 n s + 10 r s 2 / 5 - 5 n - 5 n r + 5 n s + 5 r s + n + r) A[n - 2](r + 1, -1 + s) / (n / 4 2 10 9 8 2 7 3 6 4 (n - 1 - s) ) - s (10 n - 70 n s + 210 n s - 350 n s + 350 n s 5 5 4 6 3 7 9 8 7 2 - 210 n s + 70 n s - 10 n s - 90 n + 570 n s - 1530 n s 6 3 5 4 4 5 3 6 2 7 8 + 2250 n s - 1950 n s + 990 n s - 270 n s + 30 n s + 330 n 7 6 2 5 3 4 4 3 5 - 1870 n s + 4435 n s - 5675 n s + 4200 n s - 1780 n s 2 6 7 7 6 5 2 4 3 + 395 n s - 35 n s - 630 n + 3150 n s - 6495 n s + 7095 n s 3 4 2 5 6 7 6 5 - 4380 n s + 1500 n s - 255 n s + 15 s + 660 n - 2860 n s 4 2 3 3 2 4 5 6 5 + 5020 n s - 4556 n s + 2248 n s - 568 n s + 56 s - 360 n 4 3 2 2 3 4 5 4 + 1320 n s - 1920 n s + 1392 n s - 504 n s + 72 s + 80 n 3 2 2 3 4 / 5 - 240 n s + 280 n s - 152 n s + 32 s ) A[n - 2](r, s) / (n / 4 3 8 7 7 6 (n - 1 - s) (n - 2 - s) ) + x (6 n + 10 n r - 38 n s - 70 n r s 6 2 5 2 5 3 4 3 4 4 + 98 n s + 210 n r s - 126 n s - 350 n r s + 70 n s 3 4 3 5 2 5 2 6 6 7 + 350 n r s + 14 n s - 210 n r s - 42 n s + 70 n r s + 22 n s 7 8 7 6 6 5 5 2 - 10 r s - 4 s - 42 n - 60 n r + 234 n s + 360 n r s - 522 n s 4 2 4 3 3 3 3 4 2 4 - 900 n r s + 570 n s + 1200 n r s - 270 n s - 900 n r s 2 5 5 6 6 7 6 5 - 18 n s + 360 n r s + 66 n s - 60 r s - 18 s + 117 n + 145 n r 5 4 4 2 3 2 3 3 - 557 n s - 725 n r s + 1030 n s + 1450 n r s - 890 n s 2 3 2 4 4 5 5 6 - 1450 n r s + 305 n s + 725 n r s + 23 n s - 145 r s - 28 s 5 4 4 3 3 2 2 2 - 165 n - 180 n r + 645 n s + 720 n r s - 930 n s - 1080 n r s 2 3 3 4 4 5 4 3 + 570 n s + 720 n r s - 105 n s - 180 r s - 15 s + 124 n + 124 n r 3 2 2 2 2 3 3 - 372 n s - 372 n r s + 372 n s + 372 n r s - 124 n s - 124 r s 3 2 2 2 2 2 - 48 n - 48 n r + 96 n s + 96 n r s - 48 n s - 48 r s + 8 n + 8 n r / 5 4 3 - 8 n s - 8 r s) A[n - 3](r + 1, -1 + s) / (n (n - 1 - s) (n - 2 - s) / 3 10 9 8 2 7 3 6 4 5 5 ) + s (10 n - 80 n s + 280 n s - 560 n s + 700 n s - 560 n s 4 6 3 7 2 8 9 8 7 2 + 280 n s - 80 n s + 10 n s - 120 n + 870 n s - 2730 n s 6 3 5 4 4 5 3 6 2 7 8 + 4830 n s - 5250 n s + 3570 n s - 1470 n s + 330 n s - 30 n s 8 7 6 2 5 3 4 4 + 580 n - 3770 n s + 10465 n s - 16100 n s + 14875 n s 3 5 2 6 7 8 7 6 - 8330 n s + 2695 n s - 440 n s + 25 s - 1440 n + 8280 n s 5 2 4 3 3 4 2 5 6 - 19980 n s + 26100 n s - 19800 n s + 8640 n s - 1980 n s 7 6 5 4 2 3 3 2 4 + 180 s + 1930 n - 9650 n s + 19771 n s - 21184 n s + 12476 n s 5 6 5 4 3 2 2 3 - 3814 n s + 471 s - 1320 n + 5610 n s - 9438 n s + 7854 n s 4 5 4 3 2 2 3 4 - 3234 n s + 528 s + 360 n - 1260 n s + 1656 n s - 972 n s + 216 s / 5 3 3 2 9 ) A[n - 3](r, s) / (n (n - 1 - s) (n - 2 - s) (n - 3 - s) ) - x (4 n / 8 8 7 7 2 6 2 6 3 + 10 n r - 26 n s - 80 n r s + 64 n s + 280 n r s - 56 n s 5 3 5 4 4 4 4 5 3 5 - 560 n r s - 56 n s + 700 n r s + 196 n s - 560 n r s 3 6 2 6 2 7 7 8 8 - 224 n s + 280 n r s + 136 n s - 80 n r s - 44 n s + 10 r s 9 8 7 7 6 6 2 + 6 s - 42 n - 90 n r + 246 n s + 630 n r s - 546 n s 5 2 5 3 4 3 4 4 3 4 - 1890 n r s + 462 n s + 3150 n r s + 210 n s - 3150 n r s 3 5 2 5 2 6 6 7 7 - 798 n s + 1890 n r s + 714 n s - 630 n r s - 294 n s + 90 r s 8 7 6 6 5 5 2 + 48 s + 174 n + 315 n r - 903 n s - 1890 n r s + 1764 n s 4 2 4 3 3 3 3 4 2 4 + 4725 n r s - 1365 n s - 6300 n r s - 210 n s + 4725 n r s 2 5 5 6 6 7 6 + 1071 n s - 1890 n r s - 672 n s + 315 r s + 141 s - 360 n 5 5 4 4 2 3 2 - 540 n r + 1620 n s + 2700 n r s - 2700 n s - 5400 n r s 3 3 2 3 4 5 5 6 + 1800 n s + 5400 n r s - 2700 n r s - 540 n s + 540 r s + 180 s 5 4 4 3 3 2 2 2 + 386 n + 471 n r - 1459 n s - 1884 n r s + 1976 n s + 2826 n r s 2 3 3 4 4 5 4 - 1034 n s - 1884 n r s + 46 n s + 471 r s + 85 s - 198 n 3 3 2 2 2 2 3 - 198 n r + 594 n s + 594 n r s - 594 n s - 594 n r s + 198 n s 3 3 2 2 2 2 + 198 r s + 36 n + 36 n r - 72 n s - 72 n r s + 36 n s + 36 r s ) / 5 3 3 2 A[n - 4](r + 1, -1 + s) / (n (n - 1 - s) (n - 2 - s) (n - 3 - s) ) - / 4 8 7 6 2 5 3 4 4 3 5 s (5 n - 35 n s + 105 n s - 175 n s + 175 n s - 105 n s 2 6 7 7 6 5 2 4 3 + 35 n s - 5 n s - 50 n + 310 n s - 810 n s + 1150 n s 3 4 2 5 6 7 6 5 - 950 n s + 450 n s - 110 n s + 10 s + 175 n - 945 n s 4 2 3 3 2 4 5 6 5 + 2100 n s - 2450 n s + 1575 n s - 525 n s + 70 s - 250 n 4 3 2 2 3 4 5 4 + 1150 n s - 2100 n s + 1900 n s - 850 n s + 150 s + 120 n 3 2 2 3 4 / 5 - 456 n s + 648 n s - 408 n s + 96 s ) A[n - 4](r, s) / (n / 2 2 2 8 7 (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 4 - s)) + x (n + 5 n r 7 6 6 2 5 2 5 3 4 3 - 3 n s - 35 n r s - 7 n s + 105 n r s + 49 n s - 175 n r s 4 4 3 4 3 5 2 5 2 6 6 - 105 n s + 175 n r s + 119 n s - 105 n r s - 77 n s + 35 n r s 7 7 8 7 6 6 5 + 27 n s - 5 r s - 4 s - 10 n - 40 n r + 30 n s + 240 n r s 5 2 4 2 4 3 3 3 3 4 + 30 n s - 600 n r s - 250 n s + 800 n r s + 450 n s 2 4 2 5 5 6 6 7 - 600 n r s - 390 n s + 240 n r s + 170 n s - 40 r s - 30 s 6 5 5 4 3 2 3 3 + 35 n + 105 n r - 105 n s - 525 n r s + 1050 n r s + 350 n s 2 3 2 4 4 5 5 6 - 1050 n r s - 525 n s + 525 n r s + 315 n s - 105 r s - 70 s 5 4 4 3 3 2 2 2 - 50 n - 100 n r + 150 n s + 400 n r s - 100 n s - 600 n r s 2 3 3 4 4 5 4 3 - 100 n s + 400 n r s + 150 n s - 100 r s - 50 s + 24 n + 24 n r 3 2 2 2 2 3 3 - 72 n s - 72 n r s + 72 n s + 72 n r s - 24 n s - 24 r s ) / 5 2 2 2 A[n - 5](r + 1, -1 + s) / (n (n - 1 - s) (n - 2 - s) (n - 3 - s) / 5 4 3 2 2 3 4 s (n - 4 n s + 6 n s - 4 n s + s ) A[n - 5](r, s) (n - 4 - s)) + ------------------------------------------------------- - x 5 n (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 4 - s) 4 4 3 3 2 2 2 2 3 3 4 (n r + n s - 4 n r s - 4 n s + 6 n r s + 6 n s - 4 n r s - 4 n s 4 5 / 5 + r s + s ) A[n - 6](r + 1, -1 + s) / (n (n - 1 - s) (n - 2 - s) / (n - 3 - s) (n - 4 - s)) and in Maple notation A[n](r,s) = x*(n+r)/n^5*A[n-1](r+1,-1+s)+s*(5*n^8-20*n^7*s+30*n^6*s^2-20*n^5*s^ 3+5*n^4*s^4-20*n^7+70*n^6*s-90*n^5*s^2+50*n^4*s^3-10*n^3*s^4+30*n^6-90*n^5*s+ 100*n^4*s^2-50*n^3*s^3+10*n^2*s^4-20*n^5+50*n^4*s-50*n^3*s^2+25*n^2*s^3-5*n*s^4 +5*n^4-10*n^3*s+10*n^2*s^2-5*n*s^3+s^4)/n^5/(n-1-s)^4*A[n-1](r,s)-x*(4*n^5+5*n^ 4*r-15*n^4*s-20*n^3*r*s+20*n^3*s^2+30*n^2*r*s^2-10*n^2*s^3-20*n*r*s^3+5*r*s^4+s ^5-10*n^4-10*n^3*r+30*n^3*s+30*n^2*r*s-30*n^2*s^2-30*n*r*s^2+10*n*s^3+10*r*s^3+ 10*n^3+10*n^2*r-20*n^2*s-20*n*r*s+10*n*s^2+10*r*s^2-5*n^2-5*n*r+5*n*s+5*r*s+n+r )/n^5/(n-1-s)^4*A[n-2](r+1,-1+s)-s^2*(10*n^10-70*n^9*s+210*n^8*s^2-350*n^7*s^3+ 350*n^6*s^4-210*n^5*s^5+70*n^4*s^6-10*n^3*s^7-90*n^9+570*n^8*s-1530*n^7*s^2+ 2250*n^6*s^3-1950*n^5*s^4+990*n^4*s^5-270*n^3*s^6+30*n^2*s^7+330*n^8-1870*n^7*s +4435*n^6*s^2-5675*n^5*s^3+4200*n^4*s^4-1780*n^3*s^5+395*n^2*s^6-35*n*s^7-630*n ^7+3150*n^6*s-6495*n^5*s^2+7095*n^4*s^3-4380*n^3*s^4+1500*n^2*s^5-255*n*s^6+15* s^7+660*n^6-2860*n^5*s+5020*n^4*s^2-4556*n^3*s^3+2248*n^2*s^4-568*n*s^5+56*s^6-\ 360*n^5+1320*n^4*s-1920*n^3*s^2+1392*n^2*s^3-504*n*s^4+72*s^5+80*n^4-240*n^3*s+ 280*n^2*s^2-152*n*s^3+32*s^4)/n^5/(n-1-s)^4/(n-2-s)^3*A[n-2](r,s)+x*(6*n^8+10*n ^7*r-38*n^7*s-70*n^6*r*s+98*n^6*s^2+210*n^5*r*s^2-126*n^5*s^3-350*n^4*r*s^3+70* n^4*s^4+350*n^3*r*s^4+14*n^3*s^5-210*n^2*r*s^5-42*n^2*s^6+70*n*r*s^6+22*n*s^7-\ 10*r*s^7-4*s^8-42*n^7-60*n^6*r+234*n^6*s+360*n^5*r*s-522*n^5*s^2-900*n^4*r*s^2+ 570*n^4*s^3+1200*n^3*r*s^3-270*n^3*s^4-900*n^2*r*s^4-18*n^2*s^5+360*n*r*s^5+66* n*s^6-60*r*s^6-18*s^7+117*n^6+145*n^5*r-557*n^5*s-725*n^4*r*s+1030*n^4*s^2+1450 *n^3*r*s^2-890*n^3*s^3-1450*n^2*r*s^3+305*n^2*s^4+725*n*r*s^4+23*n*s^5-145*r*s^ 5-28*s^6-165*n^5-180*n^4*r+645*n^4*s+720*n^3*r*s-930*n^3*s^2-1080*n^2*r*s^2+570 *n^2*s^3+720*n*r*s^3-105*n*s^4-180*r*s^4-15*s^5+124*n^4+124*n^3*r-372*n^3*s-372 *n^2*r*s+372*n^2*s^2+372*n*r*s^2-124*n*s^3-124*r*s^3-48*n^3-48*n^2*r+96*n^2*s+ 96*n*r*s-48*n*s^2-48*r*s^2+8*n^2+8*n*r-8*n*s-8*r*s)/n^5/(n-1-s)^4/(n-2-s)^3*A[n -3](r+1,-1+s)+s^3*(10*n^10-80*n^9*s+280*n^8*s^2-560*n^7*s^3+700*n^6*s^4-560*n^5 *s^5+280*n^4*s^6-80*n^3*s^7+10*n^2*s^8-120*n^9+870*n^8*s-2730*n^7*s^2+4830*n^6* s^3-5250*n^5*s^4+3570*n^4*s^5-1470*n^3*s^6+330*n^2*s^7-30*n*s^8+580*n^8-3770*n^ 7*s+10465*n^6*s^2-16100*n^5*s^3+14875*n^4*s^4-8330*n^3*s^5+2695*n^2*s^6-440*n*s ^7+25*s^8-1440*n^7+8280*n^6*s-19980*n^5*s^2+26100*n^4*s^3-19800*n^3*s^4+8640*n^ 2*s^5-1980*n*s^6+180*s^7+1930*n^6-9650*n^5*s+19771*n^4*s^2-21184*n^3*s^3+12476* n^2*s^4-3814*n*s^5+471*s^6-1320*n^5+5610*n^4*s-9438*n^3*s^2+7854*n^2*s^3-3234*n *s^4+528*s^5+360*n^4-1260*n^3*s+1656*n^2*s^2-972*n*s^3+216*s^4)/n^5/(n-1-s)^3/( n-2-s)^3/(n-3-s)^2*A[n-3](r,s)-x*(4*n^9+10*n^8*r-26*n^8*s-80*n^7*r*s+64*n^7*s^2 +280*n^6*r*s^2-56*n^6*s^3-560*n^5*r*s^3-56*n^5*s^4+700*n^4*r*s^4+196*n^4*s^5-\ 560*n^3*r*s^5-224*n^3*s^6+280*n^2*r*s^6+136*n^2*s^7-80*n*r*s^7-44*n*s^8+10*r*s^ 8+6*s^9-42*n^8-90*n^7*r+246*n^7*s+630*n^6*r*s-546*n^6*s^2-1890*n^5*r*s^2+462*n^ 5*s^3+3150*n^4*r*s^3+210*n^4*s^4-3150*n^3*r*s^4-798*n^3*s^5+1890*n^2*r*s^5+714* n^2*s^6-630*n*r*s^6-294*n*s^7+90*r*s^7+48*s^8+174*n^7+315*n^6*r-903*n^6*s-1890* n^5*r*s+1764*n^5*s^2+4725*n^4*r*s^2-1365*n^4*s^3-6300*n^3*r*s^3-210*n^3*s^4+ 4725*n^2*r*s^4+1071*n^2*s^5-1890*n*r*s^5-672*n*s^6+315*r*s^6+141*s^7-360*n^6-\ 540*n^5*r+1620*n^5*s+2700*n^4*r*s-2700*n^4*s^2-5400*n^3*r*s^2+1800*n^3*s^3+5400 *n^2*r*s^3-2700*n*r*s^4-540*n*s^5+540*r*s^5+180*s^6+386*n^5+471*n^4*r-1459*n^4* s-1884*n^3*r*s+1976*n^3*s^2+2826*n^2*r*s^2-1034*n^2*s^3-1884*n*r*s^3+46*n*s^4+ 471*r*s^4+85*s^5-198*n^4-198*n^3*r+594*n^3*s+594*n^2*r*s-594*n^2*s^2-594*n*r*s^ 2+198*n*s^3+198*r*s^3+36*n^3+36*n^2*r-72*n^2*s-72*n*r*s+36*n*s^2+36*r*s^2)/n^5/ (n-1-s)^3/(n-2-s)^3/(n-3-s)^2*A[n-4](r+1,-1+s)-s^4*(5*n^8-35*n^7*s+105*n^6*s^2-\ 175*n^5*s^3+175*n^4*s^4-105*n^3*s^5+35*n^2*s^6-5*n*s^7-50*n^7+310*n^6*s-810*n^5 *s^2+1150*n^4*s^3-950*n^3*s^4+450*n^2*s^5-110*n*s^6+10*s^7+175*n^6-945*n^5*s+ 2100*n^4*s^2-2450*n^3*s^3+1575*n^2*s^4-525*n*s^5+70*s^6-250*n^5+1150*n^4*s-2100 *n^3*s^2+1900*n^2*s^3-850*n*s^4+150*s^5+120*n^4-456*n^3*s+648*n^2*s^2-408*n*s^3 +96*s^4)/n^5/(n-1-s)^2/(n-2-s)^2/(n-3-s)^2/(n-4-s)*A[n-4](r,s)+x*(n^8+5*n^7*r-3 *n^7*s-35*n^6*r*s-7*n^6*s^2+105*n^5*r*s^2+49*n^5*s^3-175*n^4*r*s^3-105*n^4*s^4+ 175*n^3*r*s^4+119*n^3*s^5-105*n^2*r*s^5-77*n^2*s^6+35*n*r*s^6+27*n*s^7-5*r*s^7-\ 4*s^8-10*n^7-40*n^6*r+30*n^6*s+240*n^5*r*s+30*n^5*s^2-600*n^4*r*s^2-250*n^4*s^3 +800*n^3*r*s^3+450*n^3*s^4-600*n^2*r*s^4-390*n^2*s^5+240*n*r*s^5+170*n*s^6-40*r *s^6-30*s^7+35*n^6+105*n^5*r-105*n^5*s-525*n^4*r*s+1050*n^3*r*s^2+350*n^3*s^3-\ 1050*n^2*r*s^3-525*n^2*s^4+525*n*r*s^4+315*n*s^5-105*r*s^5-70*s^6-50*n^5-100*n^ 4*r+150*n^4*s+400*n^3*r*s-100*n^3*s^2-600*n^2*r*s^2-100*n^2*s^3+400*n*r*s^3+150 *n*s^4-100*r*s^4-50*s^5+24*n^4+24*n^3*r-72*n^3*s-72*n^2*r*s+72*n^2*s^2+72*n*r*s ^2-24*n*s^3-24*r*s^3)/n^5/(n-1-s)^2/(n-2-s)^2/(n-3-s)^2/(n-4-s)*A[n-5](r+1,-1+s )+s^5*(n^4-4*n^3*s+6*n^2*s^2-4*n*s^3+s^4)/n^5/(n-1-s)/(n-2-s)/(n-3-s)/(n-4-s)*A [n-5](r,s)-x*(n^4*r+n^4*s-4*n^3*r*s-4*n^3*s^2+6*n^2*r*s^2+6*n^2*s^3-4*n*r*s^3-4 *n*s^4+r*s^4+s^5)/n^5/(n-1-s)/(n-2-s)/(n-3-s)/(n-4-s)*A[n-6](r+1,-1+s) Proof: We claim that: Let a(n,k,r,s) be the summand on the sum defining A[n](r,s) in other words k (k - 1 + p) (n - k + q) x (r + k) (s - k) a(n, k, r, s) = ---------------------------------------- 5 (k!) (n - k)! and in Maple notation a(n,k,r,s) = 1/k!^5/(n-k)!*x^k*(r+k)^(k-1+p)*(s-k)^(n-k+q) Then the following identity is true 4 (n - s + 5) (s + r) a(n, k, r, s) 3 - ----------------------------------------------------------- + (n - s + 5) ( (n - s + 4) (n - s + 3) (n - s + 2) (n + r + 5) (n - s + 1) 5 4 3 3 2 2 2 2 3 3 n + 5 n r - 20 n r s - 10 n s + 30 n r s + 20 n s - 20 n r s 4 4 5 4 3 2 2 2 - 15 n s + 5 r s + 4 s + 15 n + 60 n r - 180 n r s - 90 n s 2 3 3 4 3 2 + 180 n r s + 120 n s - 60 r s - 45 s + 85 n + 255 n r - 510 n r s 2 2 3 2 2 - 255 n s + 255 r s + 170 s + 225 n + 450 n r - 450 r s - 225 s / 2 + 274 n + 274 r + 120) a(n + 1, k, r, s) / ((n - s + 1) (n - s + 4) / 2 2 (n + r + 5) (n - s + 3) (n - s + 2) ) 5 4 (s + 1) (n - s + 5) a(n + 1, k + 1, r - 1, s + 1) + ------------------------------------------------------------- - x (n - s + 4) (n - s + 3) (n - s + 2) (n + r + 5) (n - s + 1) 2 7 6 6 5 5 2 4 2 (n - s + 5) (4 n + 10 n r - 18 n s - 60 n r s + 24 n s + 150 n r s 4 3 3 3 3 4 2 4 2 5 5 + 10 n s - 200 n r s - 60 n s + 150 n r s + 66 n s - 60 n r s 6 6 7 6 5 5 4 - 32 n s + 10 r s + 6 s + 98 n + 210 n r - 378 n s - 1050 n r s 4 2 3 2 3 3 2 3 2 4 + 420 n s + 2100 n r s + 140 n s - 2100 n r s - 630 n s 4 5 5 6 5 4 + 1050 n r s + 462 n s - 210 r s - 112 s + 1014 n + 1815 n r 4 3 3 2 2 2 2 3 - 3255 n s - 7260 n r s + 2880 n s + 10890 n r s + 750 n s 3 4 4 5 4 3 - 7260 n r s - 2190 n s + 1815 r s + 801 s + 5740 n + 8260 n r 3 2 2 2 2 3 - 14700 n s - 24780 n r s + 9660 n s + 24780 n r s + 1820 n s 3 4 3 2 2 - 8260 r s - 2520 s + 19186 n + 20871 n r - 36687 n s - 41742 n r s 2 2 3 2 + 15816 n s + 20871 r s + 1685 s + 37842 n + 27762 n r - 47922 n s 2 - 27762 r s + 10080 s + 40756 n + 15196 r - 25560 s + 18480) / 2 3 3 a(2 + n, k, r, s) / ((n - s + 2) (n - s + 3) (n - s + 4) (n + r + 5)) / 4 5 4 3 2 2 3 4 4 - (s + 1) (5 n - 20 n s + 30 n s - 20 n s + 5 n s + 80 n 3 2 2 3 4 3 2 2 - 260 n s + 300 n s - 140 n s + 20 s + 485 n - 1200 n s + 945 n s 3 2 2 - 230 s + 1380 n - 2310 n s + 930 s + 1820 n - 1546 s + 874) 3 / 2 (n - s + 5) a(2 + n, k + 1, r - 1, s + 1) / (x (n - s + 1) (n - s + 4) / 2 2 7 6 (n - s + 3) (n + r + 5) (n - s + 2) ) + (n - s + 5) (6 n + 10 n r 6 5 5 2 4 2 4 3 3 3 - 32 n s - 60 n r s + 66 n s + 150 n r s - 60 n s - 200 n r s 3 4 2 4 2 5 5 6 6 7 + 10 n s + 150 n r s + 24 n s - 60 n r s - 18 n s + 10 r s + 4 s 6 5 5 4 4 2 3 2 + 168 n + 240 n r - 768 n s - 1200 n r s + 1320 n s + 2400 n r s 3 3 2 3 2 4 4 5 5 - 960 n s - 2400 n r s + 120 n s + 1200 n r s + 192 n s - 240 r s 6 5 4 4 3 3 2 - 72 s + 2007 n + 2395 n r - 7640 n s - 9580 n r s + 10490 n s 2 2 2 3 3 4 4 5 + 14370 n r s - 5700 n s - 9580 n r s + 455 n s + 2395 r s + 388 s 4 3 3 2 2 2 + 13260 n + 12720 n r - 40320 n s - 38160 n r s + 41400 n s 2 3 3 4 3 2 + 38160 n r s - 14880 n s - 12720 r s + 540 s + 52324 n + 37924 n r 2 2 2 3 - 119048 n s - 75848 n r s + 81124 n s + 37924 r s - 14400 s 2 2 + 123312 n + 60192 n r - 186432 n s - 60192 r s + 63120 s + 160703 n / 4 + 39743 r - 120960 s + 89340) a(n + 3, k, r, s) / ((n - s + 4) / 3 3 8 7 6 2 (n + r + 5) (n - s + 3) ) + (s + 1) (10 n - 60 n s + 150 n s 5 3 4 4 3 5 2 6 7 6 - 200 n s + 150 n s - 60 n s + 10 n s + 300 n - 1590 n s 5 2 4 3 3 4 2 5 6 6 + 3450 n s - 3900 n s + 2400 n s - 750 n s + 90 n s + 3880 n 5 4 2 3 3 2 4 5 - 17790 n s + 32565 n s - 29960 n s + 14190 n s - 3090 n s 6 5 4 3 2 2 3 + 205 s + 28240 n - 108870 n s + 161340 n s - 113200 n s 4 5 4 3 2 2 + 36660 n s - 4170 s + 126445 n - 393330 n s + 442191 n s 3 4 3 2 2 - 210172 n s + 34866 s + 356470 n - 838458 n s + 635268 n s 3 2 2 - 153280 s + 617641 n - 975978 n s + 373533 s + 601142 n - 478350 s 2 / 2 + 251596) (n - s + 5) a(n + 3, k + 1, r - 1, s + 1) / (x (n - s + 2) / 3 3 5 4 4 (n + r + 5) (n - s + 4) (n - s + 3) ) - (4 n + 5 n r - 15 n s 3 3 2 2 2 2 3 3 4 5 - 20 n r s + 20 n s + 30 n r s - 10 n s - 20 n r s + 5 r s + s 4 3 3 2 2 2 2 + 90 n + 90 n r - 270 n s - 270 n r s + 270 n s + 270 n r s 3 3 3 2 2 - 90 n s - 90 r s + 810 n + 610 n r - 1820 n s - 1220 n r s 2 2 3 2 + 1210 n s + 610 r s - 200 s + 3645 n + 1845 n r - 5445 n s - 1845 r s 2 / + 1800 s + 8201 n + 2101 r - 6100 s + 7380) a(4 + n, k, r, s) / ( / 4 2 9 8 (n - s + 4) (n + r + 5)) - (s + 1) (n - s + 5) (10 n - 60 n s 7 2 6 3 5 4 4 5 3 6 8 + 150 n s - 200 n s + 150 n s - 60 n s + 10 n s + 390 n 7 6 2 5 3 4 4 3 5 - 2100 n s + 4650 n s - 5400 n s + 3450 n s - 1140 n s 2 6 7 6 5 2 4 3 + 150 n s + 6720 n - 31960 n s + 61395 n s - 60380 n s 3 4 2 5 6 6 5 + 31570 n s - 8100 n s + 755 n s + 67140 n - 276210 n s 4 2 3 3 2 4 5 6 + 447435 n s - 357720 n s + 143550 n s - 25470 n s + 1275 s 5 4 3 2 2 3 4 + 428625 n - 1482490 n s + 1943490 n s - 1183856 n s + 324077 n s 5 4 3 2 2 3 - 29846 s + 1813185 n - 5059820 n s + 5030682 n s - 2074452 n s 4 3 2 2 3 + 290405 s + 5082490 n - 10723948 n s + 7184517 n s - 1503316 s 2 2 + 9103314 n - 12904222 n s + 4366837 s + 9454507 n - 6749878 s + 4338443 / 3 4 ) a(4 + n, k + 1, r - 1, s + 1) / (x (n - s + 3) (n - s + 4) / 8 7 6 2 (n + r + 5)) + a(n + 5, k, r, s) + (s + 1) (5 n - 20 n s + 30 n s 5 3 4 4 7 6 5 2 4 3 - 20 n s + 5 n s + 200 n - 710 n s + 930 n s - 530 n s 3 4 6 5 4 2 3 3 2 4 + 110 n s + 3490 n - 10770 n s + 11980 n s - 5610 n s + 910 n s 5 4 3 2 2 3 4 + 34700 n - 90480 n s + 82060 n s - 29635 n s + 3355 n s 4 3 2 2 3 4 + 215010 n - 454620 n s + 315145 n s - 78085 n s + 4651 s 3 2 2 3 2 + 850200 n - 1366065 n s + 643215 n s - 82046 s + 2095245 n 2 - 2272895 n s + 544956 s + 2942450 n - 1615346 s + 1803001) / 4 a(n + 5, k + 1, r - 1, s + 1) / (x (n - s + 4) (n + r + 5)) / 5 (n + 6) a(n + 6, k + 1, r - 1, s + 1) - -------------------------------------- = 0 x (n + r + 5) and in Maple notation: -(n-s+5)^4*(s+r)/(n-s+4)/(n-s+3)/(n-s+2)/(n+r+5)/(n-s+1)*a(n,k,r,s)+(n-s+5)^3*( n^5+5*n^4*r-20*n^3*r*s-10*n^3*s^2+30*n^2*r*s^2+20*n^2*s^3-20*n*r*s^3-15*n*s^4+5 *r*s^4+4*s^5+15*n^4+60*n^3*r-180*n^2*r*s-90*n^2*s^2+180*n*r*s^2+120*n*s^3-60*r* s^3-45*s^4+85*n^3+255*n^2*r-510*n*r*s-255*n*s^2+255*r*s^2+170*s^3+225*n^2+450*n *r-450*r*s-225*s^2+274*n+274*r+120)/(n-s+1)/(n-s+4)^2/(n+r+5)/(n-s+3)^2/(n-s+2) ^2*a(n+1,k,r,s)+(s+1)^5*(n-s+5)^4/x/(n-s+4)/(n-s+3)/(n-s+2)/(n+r+5)/(n-s+1)*a(n +1,k+1,r-1,s+1)-(n-s+5)^2*(4*n^7+10*n^6*r-18*n^6*s-60*n^5*r*s+24*n^5*s^2+150*n^ 4*r*s^2+10*n^4*s^3-200*n^3*r*s^3-60*n^3*s^4+150*n^2*r*s^4+66*n^2*s^5-60*n*r*s^5 -32*n*s^6+10*r*s^6+6*s^7+98*n^6+210*n^5*r-378*n^5*s-1050*n^4*r*s+420*n^4*s^2+ 2100*n^3*r*s^2+140*n^3*s^3-2100*n^2*r*s^3-630*n^2*s^4+1050*n*r*s^4+462*n*s^5-\ 210*r*s^5-112*s^6+1014*n^5+1815*n^4*r-3255*n^4*s-7260*n^3*r*s+2880*n^3*s^2+ 10890*n^2*r*s^2+750*n^2*s^3-7260*n*r*s^3-2190*n*s^4+1815*r*s^4+801*s^5+5740*n^4 +8260*n^3*r-14700*n^3*s-24780*n^2*r*s+9660*n^2*s^2+24780*n*r*s^2+1820*n*s^3-\ 8260*r*s^3-2520*s^4+19186*n^3+20871*n^2*r-36687*n^2*s-41742*n*r*s+15816*n*s^2+ 20871*r*s^2+1685*s^3+37842*n^2+27762*n*r-47922*n*s-27762*r*s+10080*s^2+40756*n+ 15196*r-25560*s+18480)/(n-s+2)^2/(n-s+3)^3/(n-s+4)^3/(n+r+5)*a(2+n,k,r,s)-(s+1) ^4*(5*n^5-20*n^4*s+30*n^3*s^2-20*n^2*s^3+5*n*s^4+80*n^4-260*n^3*s+300*n^2*s^2-\ 140*n*s^3+20*s^4+485*n^3-1200*n^2*s+945*n*s^2-230*s^3+1380*n^2-2310*n*s+930*s^2 +1820*n-1546*s+874)*(n-s+5)^3/x/(n-s+1)/(n-s+4)^2/(n-s+3)^2/(n+r+5)/(n-s+2)^2*a (2+n,k+1,r-1,s+1)+(n-s+5)*(6*n^7+10*n^6*r-32*n^6*s-60*n^5*r*s+66*n^5*s^2+150*n^ 4*r*s^2-60*n^4*s^3-200*n^3*r*s^3+10*n^3*s^4+150*n^2*r*s^4+24*n^2*s^5-60*n*r*s^5 -18*n*s^6+10*r*s^6+4*s^7+168*n^6+240*n^5*r-768*n^5*s-1200*n^4*r*s+1320*n^4*s^2+ 2400*n^3*r*s^2-960*n^3*s^3-2400*n^2*r*s^3+120*n^2*s^4+1200*n*r*s^4+192*n*s^5-\ 240*r*s^5-72*s^6+2007*n^5+2395*n^4*r-7640*n^4*s-9580*n^3*r*s+10490*n^3*s^2+ 14370*n^2*r*s^2-5700*n^2*s^3-9580*n*r*s^3+455*n*s^4+2395*r*s^4+388*s^5+13260*n^ 4+12720*n^3*r-40320*n^3*s-38160*n^2*r*s+41400*n^2*s^2+38160*n*r*s^2-14880*n*s^3 -12720*r*s^3+540*s^4+52324*n^3+37924*n^2*r-119048*n^2*s-75848*n*r*s+81124*n*s^2 +37924*r*s^2-14400*s^3+123312*n^2+60192*n*r-186432*n*s-60192*r*s+63120*s^2+ 160703*n+39743*r-120960*s+89340)/(n-s+4)^4/(n+r+5)/(n-s+3)^3*a(n+3,k,r,s)+(s+1) ^3*(10*n^8-60*n^7*s+150*n^6*s^2-200*n^5*s^3+150*n^4*s^4-60*n^3*s^5+10*n^2*s^6+ 300*n^7-1590*n^6*s+3450*n^5*s^2-3900*n^4*s^3+2400*n^3*s^4-750*n^2*s^5+90*n*s^6+ 3880*n^6-17790*n^5*s+32565*n^4*s^2-29960*n^3*s^3+14190*n^2*s^4-3090*n*s^5+205*s ^6+28240*n^5-108870*n^4*s+161340*n^3*s^2-113200*n^2*s^3+36660*n*s^4-4170*s^5+ 126445*n^4-393330*n^3*s+442191*n^2*s^2-210172*n*s^3+34866*s^4+356470*n^3-838458 *n^2*s+635268*n*s^2-153280*s^3+617641*n^2-975978*n*s+373533*s^2+601142*n-478350 *s+251596)*(n-s+5)^2/x/(n-s+2)^2/(n+r+5)/(n-s+4)^3/(n-s+3)^3*a(n+3,k+1,r-1,s+1) -(4*n^5+5*n^4*r-15*n^4*s-20*n^3*r*s+20*n^3*s^2+30*n^2*r*s^2-10*n^2*s^3-20*n*r*s ^3+5*r*s^4+s^5+90*n^4+90*n^3*r-270*n^3*s-270*n^2*r*s+270*n^2*s^2+270*n*r*s^2-90 *n*s^3-90*r*s^3+810*n^3+610*n^2*r-1820*n^2*s-1220*n*r*s+1210*n*s^2+610*r*s^2-\ 200*s^3+3645*n^2+1845*n*r-5445*n*s-1845*r*s+1800*s^2+8201*n+2101*r-6100*s+7380) /(n-s+4)^4/(n+r+5)*a(4+n,k,r,s)-(s+1)^2*(n-s+5)*(10*n^9-60*n^8*s+150*n^7*s^2-\ 200*n^6*s^3+150*n^5*s^4-60*n^4*s^5+10*n^3*s^6+390*n^8-2100*n^7*s+4650*n^6*s^2-\ 5400*n^5*s^3+3450*n^4*s^4-1140*n^3*s^5+150*n^2*s^6+6720*n^7-31960*n^6*s+61395*n ^5*s^2-60380*n^4*s^3+31570*n^3*s^4-8100*n^2*s^5+755*n*s^6+67140*n^6-276210*n^5* s+447435*n^4*s^2-357720*n^3*s^3+143550*n^2*s^4-25470*n*s^5+1275*s^6+428625*n^5-\ 1482490*n^4*s+1943490*n^3*s^2-1183856*n^2*s^3+324077*n*s^4-29846*s^5+1813185*n^ 4-5059820*n^3*s+5030682*n^2*s^2-2074452*n*s^3+290405*s^4+5082490*n^3-10723948*n ^2*s+7184517*n*s^2-1503316*s^3+9103314*n^2-12904222*n*s+4366837*s^2+9454507*n-\ 6749878*s+4338443)/x/(n-s+3)^3/(n-s+4)^4/(n+r+5)*a(4+n,k+1,r-1,s+1)+a(n+5,k,r,s )+(s+1)*(5*n^8-20*n^7*s+30*n^6*s^2-20*n^5*s^3+5*n^4*s^4+200*n^7-710*n^6*s+930*n ^5*s^2-530*n^4*s^3+110*n^3*s^4+3490*n^6-10770*n^5*s+11980*n^4*s^2-5610*n^3*s^3+ 910*n^2*s^4+34700*n^5-90480*n^4*s+82060*n^3*s^2-29635*n^2*s^3+3355*n*s^4+215010 *n^4-454620*n^3*s+315145*n^2*s^2-78085*n*s^3+4651*s^4+850200*n^3-1366065*n^2*s+ 643215*n*s^2-82046*s^3+2095245*n^2-2272895*n*s+544956*s^2+2942450*n-1615346*s+ 1803001)/x/(n-s+4)^4/(n+r+5)*a(n+5,k+1,r-1,s+1)-(n+6)^5/x/(n+r+5)*a(n+6,k+1,r-1 ,s+1) = 0 The proof of this identity is routine (divide by a(n,k,r,s), simplify each t\ erm,and now each term is a rational function. Now add them all up and \ verify that they add up to zero.) Now sum it from k=0 to k=n, which is the same as from k=-infinity to k=infin\ ity (since it vanishes for k<0 and k>n 4 (n - s + 5) (s + r) A[n](r, s) 3 - ----------------------------------------------------------- + (n - s + 5) ( (n - s + 4) (n - s + 3) (n - s + 2) (n + r + 5) (n - s + 1) 5 4 3 3 2 2 2 2 3 3 n + 5 n r - 20 n r s - 10 n s + 30 n r s + 20 n s - 20 n r s 4 4 5 4 3 2 2 2 - 15 n s + 5 r s + 4 s + 15 n + 60 n r - 180 n r s - 90 n s 2 3 3 4 3 2 + 180 n r s + 120 n s - 60 r s - 45 s + 85 n + 255 n r - 510 n r s 2 2 3 2 2 - 255 n s + 255 r s + 170 s + 225 n + 450 n r - 450 r s - 225 s / 2 + 274 n + 274 r + 120) A[n + 1](r, s) / ((n - s + 1) (n - s + 4) / 2 2 (n + r + 5) (n - s + 3) (n - s + 2) ) 5 4 (s + 1) (n - s + 5) A[n + 1](r - 1, s + 1) + ------------------------------------------------------------- - x (n - s + 4) (n - s + 3) (n - s + 2) (n + r + 5) (n - s + 1) 2 7 6 6 5 5 2 4 2 (n - s + 5) (4 n + 10 n r - 18 n s - 60 n r s + 24 n s + 150 n r s 4 3 3 3 3 4 2 4 2 5 5 + 10 n s - 200 n r s - 60 n s + 150 n r s + 66 n s - 60 n r s 6 6 7 6 5 5 4 - 32 n s + 10 r s + 6 s + 98 n + 210 n r - 378 n s - 1050 n r s 4 2 3 2 3 3 2 3 2 4 + 420 n s + 2100 n r s + 140 n s - 2100 n r s - 630 n s 4 5 5 6 5 4 + 1050 n r s + 462 n s - 210 r s - 112 s + 1014 n + 1815 n r 4 3 3 2 2 2 2 3 - 3255 n s - 7260 n r s + 2880 n s + 10890 n r s + 750 n s 3 4 4 5 4 3 - 7260 n r s - 2190 n s + 1815 r s + 801 s + 5740 n + 8260 n r 3 2 2 2 2 3 - 14700 n s - 24780 n r s + 9660 n s + 24780 n r s + 1820 n s 3 4 3 2 2 - 8260 r s - 2520 s + 19186 n + 20871 n r - 36687 n s - 41742 n r s 2 2 3 2 + 15816 n s + 20871 r s + 1685 s + 37842 n + 27762 n r - 47922 n s 2 - 27762 r s + 10080 s + 40756 n + 15196 r - 25560 s + 18480) / 2 3 3 A[2 + n](r, s) / ((n - s + 2) (n - s + 3) (n - s + 4) (n + r + 5)) - / 4 5 4 3 2 2 3 4 4 3 (s + 1) (5 n - 20 n s + 30 n s - 20 n s + 5 n s + 80 n - 260 n s 2 2 3 4 3 2 2 3 + 300 n s - 140 n s + 20 s + 485 n - 1200 n s + 945 n s - 230 s 2 2 3 + 1380 n - 2310 n s + 930 s + 1820 n - 1546 s + 874) (n - s + 5) / 2 2 A[2 + n](r - 1, s + 1) / (x (n - s + 1) (n - s + 4) (n - s + 3) / 2 7 6 6 (n + r + 5) (n - s + 2) ) + (n - s + 5) (6 n + 10 n r - 32 n s 5 5 2 4 2 4 3 3 3 3 4 - 60 n r s + 66 n s + 150 n r s - 60 n s - 200 n r s + 10 n s 2 4 2 5 5 6 6 7 6 + 150 n r s + 24 n s - 60 n r s - 18 n s + 10 r s + 4 s + 168 n 5 5 4 4 2 3 2 + 240 n r - 768 n s - 1200 n r s + 1320 n s + 2400 n r s 3 3 2 3 2 4 4 5 5 - 960 n s - 2400 n r s + 120 n s + 1200 n r s + 192 n s - 240 r s 6 5 4 4 3 3 2 - 72 s + 2007 n + 2395 n r - 7640 n s - 9580 n r s + 10490 n s 2 2 2 3 3 4 4 5 + 14370 n r s - 5700 n s - 9580 n r s + 455 n s + 2395 r s + 388 s 4 3 3 2 2 2 + 13260 n + 12720 n r - 40320 n s - 38160 n r s + 41400 n s 2 3 3 4 3 2 + 38160 n r s - 14880 n s - 12720 r s + 540 s + 52324 n + 37924 n r 2 2 2 3 - 119048 n s - 75848 n r s + 81124 n s + 37924 r s - 14400 s 2 2 + 123312 n + 60192 n r - 186432 n s - 60192 r s + 63120 s + 160703 n / 4 + 39743 r - 120960 s + 89340) A[n + 3](r, s) / ((n - s + 4) (n + r + 5) / 3 3 8 7 6 2 5 3 (n - s + 3) ) + (s + 1) (10 n - 60 n s + 150 n s - 200 n s 4 4 3 5 2 6 7 6 5 2 + 150 n s - 60 n s + 10 n s + 300 n - 1590 n s + 3450 n s 4 3 3 4 2 5 6 6 5 - 3900 n s + 2400 n s - 750 n s + 90 n s + 3880 n - 17790 n s 4 2 3 3 2 4 5 6 5 + 32565 n s - 29960 n s + 14190 n s - 3090 n s + 205 s + 28240 n 4 3 2 2 3 4 5 - 108870 n s + 161340 n s - 113200 n s + 36660 n s - 4170 s 4 3 2 2 3 4 + 126445 n - 393330 n s + 442191 n s - 210172 n s + 34866 s 3 2 2 3 2 + 356470 n - 838458 n s + 635268 n s - 153280 s + 617641 n 2 2 - 975978 n s + 373533 s + 601142 n - 478350 s + 251596) (n - s + 5) / 2 3 A[n + 3](r - 1, s + 1) / (x (n - s + 2) (n + r + 5) (n - s + 4) / 3 5 4 4 3 3 2 (n - s + 3) ) - (4 n + 5 n r - 15 n s - 20 n r s + 20 n s 2 2 2 3 3 4 5 4 3 + 30 n r s - 10 n s - 20 n r s + 5 r s + s + 90 n + 90 n r 3 2 2 2 2 3 3 - 270 n s - 270 n r s + 270 n s + 270 n r s - 90 n s - 90 r s 3 2 2 2 2 + 810 n + 610 n r - 1820 n s - 1220 n r s + 1210 n s + 610 r s 3 2 2 - 200 s + 3645 n + 1845 n r - 5445 n s - 1845 r s + 1800 s + 8201 n / 4 + 2101 r - 6100 s + 7380) A[4 + n](r, s) / ((n - s + 4) (n + r + 5)) - / 2 9 8 7 2 6 3 5 4 (s + 1) (n - s + 5) (10 n - 60 n s + 150 n s - 200 n s + 150 n s 4 5 3 6 8 7 6 2 5 3 - 60 n s + 10 n s + 390 n - 2100 n s + 4650 n s - 5400 n s 4 4 3 5 2 6 7 6 5 2 + 3450 n s - 1140 n s + 150 n s + 6720 n - 31960 n s + 61395 n s 4 3 3 4 2 5 6 6 - 60380 n s + 31570 n s - 8100 n s + 755 n s + 67140 n 5 4 2 3 3 2 4 5 - 276210 n s + 447435 n s - 357720 n s + 143550 n s - 25470 n s 6 5 4 3 2 2 3 + 1275 s + 428625 n - 1482490 n s + 1943490 n s - 1183856 n s 4 5 4 3 2 2 + 324077 n s - 29846 s + 1813185 n - 5059820 n s + 5030682 n s 3 4 3 2 2 - 2074452 n s + 290405 s + 5082490 n - 10723948 n s + 7184517 n s 3 2 2 - 1503316 s + 9103314 n - 12904222 n s + 4366837 s + 9454507 n / 3 - 6749878 s + 4338443) A[4 + n](r - 1, s + 1) / (x (n - s + 3) / 4 8 7 (n - s + 4) (n + r + 5)) + A[n + 5](r, s) + (s + 1) (5 n - 20 n s 6 2 5 3 4 4 7 6 5 2 + 30 n s - 20 n s + 5 n s + 200 n - 710 n s + 930 n s 4 3 3 4 6 5 4 2 3 3 - 530 n s + 110 n s + 3490 n - 10770 n s + 11980 n s - 5610 n s 2 4 5 4 3 2 2 3 + 910 n s + 34700 n - 90480 n s + 82060 n s - 29635 n s 4 4 3 2 2 3 + 3355 n s + 215010 n - 454620 n s + 315145 n s - 78085 n s 4 3 2 2 3 2 + 4651 s + 850200 n - 1366065 n s + 643215 n s - 82046 s + 2095245 n 2 - 2272895 n s + 544956 s + 2942450 n - 1615346 s + 1803001) / 4 A[n + 5](r - 1, s + 1) / (x (n - s + 4) (n + r + 5)) / 5 (n + 6) A[n + 6](r - 1, s + 1) - ------------------------------- = 0 x (n + r + 5) replacing n by, n - 6, changing variables, and moving a[n](r,s) to the left \ side, yields the statement of the thereom. QED. ------------------------------------------------- This took, 9.785, seconds. Theorem: For any integers p and q define the Abel-sum type sequence by n ----- k (k - 1 + p) (n - k + q) \ x (r + k) (s - k) A[n](r, s) = ) ---------------------------------------- / 6 ----- (k!) (n - k)! k = 0 Then we have the following functional recurrence expressing A[n](r.s) in ter\ ms of , A[n - 1](r, s), A[n - 2](r, s), A[n - 3](r, s), A[n - 4](r, s), A[n - 5](r, s), A[n - 6](r, s), A[n - 7](r, s) : x (n + r) A[n - 1](r + 1, s - 1) 10 9 8 2 A[n](r, s) = -------------------------------- + s (6 n - 30 n s + 60 n s 6 n 7 3 6 4 5 5 9 8 7 2 6 3 - 60 n s + 30 n s - 6 n s - 30 n + 135 n s - 240 n s + 210 n s 5 4 4 5 8 7 6 2 5 3 - 90 n s + 15 n s + 60 n - 240 n s + 380 n s - 300 n s 4 4 3 5 7 6 5 2 4 3 + 120 n s - 20 n s - 60 n + 210 n s - 300 n s + 225 n s 3 4 2 5 6 5 4 2 3 3 2 4 - 90 n s + 15 n s + 30 n - 90 n s + 120 n s - 90 n s + 36 n s 5 5 4 3 2 2 3 4 5 - 6 n s - 6 n + 15 n s - 20 n s + 15 n s - 6 n s + s ) / 6 5 6 5 5 A[n - 1](r, s) / (n (n - 1 - s) ) - x (5 n + 6 n r - 24 n s / 4 4 2 3 2 3 3 2 3 2 4 - 30 n r s + 45 n s + 60 n r s - 40 n s - 60 n r s + 15 n s 4 5 6 5 4 4 3 + 30 n r s - 6 r s - s - 15 n - 15 n r + 60 n s + 60 n r s 3 2 2 2 2 3 3 4 4 4 - 90 n s - 90 n r s + 60 n s + 60 n r s - 15 n s - 15 r s + 20 n 3 3 2 2 2 2 3 3 + 20 n r - 60 n s - 60 n r s + 60 n s + 60 n r s - 20 n s - 20 r s 3 2 2 2 2 2 - 15 n - 15 n r + 30 n s + 30 n r s - 15 n s - 15 r s + 6 n + 6 n r / 6 5 2 - 6 n s - 6 r s - n - r) A[n - 2](r + 1, s - 1) / (n (n - 1 - s) ) - s / 13 12 11 2 10 3 9 4 8 5 (15 n - 135 n s + 540 n s - 1260 n s + 1890 n s - 1890 n s 7 6 6 7 5 8 4 9 12 11 + 1260 n s - 540 n s + 135 n s - 15 n s - 180 n + 1500 n s 10 2 9 3 8 4 7 5 6 6 - 5520 n s + 11760 n s - 15960 n s + 14280 n s - 8400 n s 5 7 4 8 3 9 11 10 9 2 + 3120 n s - 660 n s + 60 n s + 930 n - 7130 n s + 23975 n s 8 3 7 4 6 5 5 6 4 7 - 46305 n s + 56455 n s - 44905 n s + 23205 n s - 7475 n s 3 8 2 9 10 9 8 2 + 1355 n s - 105 n s - 2700 n + 18900 n s - 57600 n s 7 3 6 4 5 5 4 6 3 7 + 99990 n s - 108540 n s + 76050 n s - 34200 n s + 9450 n s 2 8 9 9 8 7 2 6 3 - 1440 n s + 90 n s + 4815 n - 30495 n s + 83415 n s - 128781 n s 5 4 4 5 3 6 2 7 8 9 + 122986 n s - 74780 n s + 28615 n s - 6505 n s + 761 n s - 31 s 8 7 6 2 5 3 4 4 - 5400 n + 30600 n s - 74250 n s + 100620 n s - 83130 n s 3 5 2 6 7 8 7 6 + 42720 n s - 13230 n s + 2220 n s - 150 s + 3720 n - 18600 n s 5 2 4 3 3 4 2 5 6 + 39480 n s - 46200 n s + 32200 n s - 13320 n s + 3000 n s 7 6 5 4 2 3 3 2 4 - 280 s - 1440 n + 6240 n s - 11400 n s + 11280 n s - 6360 n s 5 6 5 4 3 2 2 3 + 1920 n s - 240 s + 240 n - 880 n s + 1360 n s - 1104 n s 4 5 / 6 5 4 + 464 n s - 80 s ) A[n - 2](r, s) / (n (n - 1 - s) (n - 2 - s) ) + x / 10 9 9 8 8 2 7 2 (10 n + 15 n r - 85 n s - 135 n r s + 315 n s + 540 n r s 7 3 6 3 6 4 5 4 5 5 - 660 n s - 1260 n r s + 840 n s + 1890 n r s - 630 n s 4 5 4 6 3 6 3 7 2 7 - 1890 n r s + 210 n s + 1260 n r s + 60 n s - 540 n r s 2 8 8 9 9 10 9 8 - 90 n s + 135 n r s + 35 n s - 15 r s - 5 s - 90 n - 120 n r 8 7 7 2 6 2 6 3 + 690 n s + 960 n r s - 2280 n s - 3360 n r s + 4200 n s 5 3 5 4 4 4 4 5 3 5 + 6720 n r s - 4620 n s - 8400 n r s + 2940 n s + 6720 n r s 3 6 2 6 2 7 7 8 8 - 840 n s - 3360 n r s - 120 n s + 960 n r s + 150 n s - 120 r s 9 8 7 7 6 6 2 - 30 s + 345 n + 415 n r - 2345 n s - 2905 n r s + 6755 n s 5 2 5 3 4 3 4 4 3 4 + 8715 n r s - 10605 n s - 14525 n r s + 9625 n s + 14525 n r s 3 5 2 5 2 6 6 7 - 4795 n s - 8715 n r s + 945 n s + 2905 n r s + 145 n s 7 8 7 6 6 5 - 415 r s - 70 s - 735 n - 810 n r + 4335 n s + 4860 n r s 5 2 4 2 4 3 3 3 3 4 - 10575 n s - 12150 n r s + 13575 n s + 16200 n r s - 9525 n s 2 4 2 5 5 6 6 7 - 12150 n r s + 3285 n s + 4860 n r s - 285 n s - 810 r s - 75 s 6 5 5 4 4 2 3 2 + 954 n + 985 n r - 4739 n s - 4925 n r s + 9385 n s + 9850 n r s 3 3 2 3 2 4 4 5 - 9230 n s - 9850 n r s + 4460 n s + 4925 n r s - 799 n s 5 6 5 4 4 3 - 985 r s - 31 s - 780 n - 780 n r + 3120 n s + 3120 n r s 3 2 2 2 2 3 3 4 - 4680 n s - 4680 n r s + 3120 n s + 3120 n r s - 780 n s 4 4 3 3 2 2 2 - 780 r s + 400 n + 400 n r - 1200 n s - 1200 n r s + 1200 n s 2 3 3 3 2 2 + 1200 n r s - 400 n s - 400 r s - 120 n - 120 n r + 240 n s 2 2 2 + 240 n r s - 120 n s - 120 r s + 16 n + 16 n r - 16 n s - 16 r s) / 6 5 4 3 14 A[n - 3](r + 1, s - 1) / (n (n - 1 - s) (n - 2 - s) ) + 2 s (10 n / 13 12 2 11 3 10 4 9 5 - 110 n s + 550 n s - 1650 n s + 3300 n s - 4620 n s 8 6 7 7 6 8 5 9 4 10 + 4620 n s - 3300 n s + 1650 n s - 550 n s + 110 n s 3 11 13 12 11 2 10 3 - 10 n s - 180 n + 1845 n s - 8550 n s + 23625 n s 9 4 8 5 7 6 6 7 5 8 - 43200 n s + 54810 n s - 49140 n s + 31050 n s - 13500 n s 4 9 3 10 2 11 12 11 + 3825 n s - 630 n s + 45 n s + 1410 n - 13395 n s 10 2 9 3 8 4 7 5 6 6 + 57180 n s - 144495 n s + 239580 n s - 272790 n s + 216720 n s 5 7 4 8 3 9 2 10 11 - 119430 n s + 44370 n s - 10455 n s + 1380 n s - 75 n s 11 10 9 2 8 3 7 4 - 6300 n + 55125 n s - 215190 n s + 493065 n s - 733680 n s 6 5 5 6 4 7 3 8 2 9 + 740250 n s - 512820 n s + 241290 n s - 74340 n s + 13905 n s 10 11 10 9 8 2 - 1350 n s + 45 s + 17670 n - 141360 n s + 500238 n s 7 3 6 4 5 5 4 6 - 1028344 n s + 1355284 n s - 1191288 n s + 703220 n s 3 7 2 8 9 10 9 - 273128 n s + 66054 n s - 8824 n s + 478 s - 32220 n 8 7 2 6 3 5 4 + 233595 n s - 741402 n s + 1349319 n s - 1548072 n s 4 5 3 6 2 7 8 9 + 1157715 n s - 562290 n s + 170217 n s - 28944 n s + 2082 s 8 7 6 2 5 3 4 4 + 38150 n - 247975 n s + 696560 n s - 1103175 n s + 1076100 n s 3 5 2 6 7 8 7 - 661125 n s + 249400 n s - 52685 n s + 4750 s - 28260 n 6 5 2 4 3 3 4 2 5 + 162495 n s - 397170 n s + 534915 n s - 428760 n s + 204525 n s 6 7 6 5 4 2 - 53730 n s + 5985 s + 11880 n - 59400 n s + 123552 n s 3 3 2 4 5 6 5 4 - 137016 n s + 85536 n s - 28512 n s + 3960 s - 2160 n + 9180 n s 3 2 2 3 4 5 / 6 - 15768 n s + 13716 n s - 6048 n s + 1080 s ) A[n - 3](r, s) / (n / 4 4 3 12 11 11 (n - 1 - s) (n - 2 - s) (n - 3 - s) ) - x (10 n + 20 n r - 100 n s 10 10 2 9 2 9 3 8 3 - 220 n r s + 440 n s + 1100 n r s - 1100 n s - 3300 n r s 8 4 7 4 7 5 6 5 5 6 + 1650 n s + 6600 n r s - 1320 n s - 9240 n r s + 9240 n r s 5 7 4 7 4 8 3 8 3 9 + 1320 n s - 6600 n r s - 1650 n s + 3300 n r s + 1100 n s 2 9 2 10 10 11 11 12 - 1100 n r s - 440 n s + 220 n r s + 100 n s - 20 r s - 10 s 11 10 10 9 9 2 - 150 n - 270 n r + 1380 n s + 2700 n r s - 5550 n s 8 2 8 3 7 3 7 4 - 12150 n r s + 12600 n s + 32400 n r s - 17100 n s 6 4 6 5 5 5 5 6 4 6 - 56700 n r s + 12600 n s + 68040 n r s - 1260 n s - 56700 n r s 4 7 3 7 3 8 2 8 2 9 - 7200 n s + 32400 n r s + 7650 n s - 12150 n r s - 3900 n s 9 10 10 11 10 9 + 2700 n r s + 1050 n s - 270 r s - 120 s + 965 n + 1560 n r 9 8 8 2 7 2 7 3 - 8090 n s - 14040 n r s + 29385 n s + 56160 n r s - 59640 n s 6 3 6 4 5 4 5 5 - 131040 n r s + 71610 n s + 196560 n r s - 46620 n s 4 5 4 6 3 6 3 7 - 196560 n r s + 6090 n s + 131040 n r s + 15240 n s 2 7 2 8 8 9 9 - 56160 n r s - 12735 n s + 14040 n r s + 4390 n s - 1560 r s 10 9 8 8 7 7 2 - 595 s - 3480 n - 5040 n r + 26280 n s + 40320 n r s - 84960 n s 6 2 6 3 5 3 5 4 - 141120 n r s + 151200 n s + 282240 n r s - 156240 n s 4 4 4 5 3 5 3 6 - 352800 n r s + 85680 n s + 282240 n r s - 10080 n s 2 6 2 7 7 8 8 - 141120 n r s - 15840 n s + 40320 n r s + 9000 n s - 5040 r s 9 8 7 7 6 - 1560 s + 7716 n + 10000 n r - 51728 n s - 70000 n r s 6 2 5 2 5 3 4 3 + 146048 n s + 210000 n r s - 222096 n s - 350000 n r s 4 4 3 4 3 5 2 5 + 190120 n s + 350000 n r s - 82096 n s - 210000 n r s 2 6 6 7 7 8 7 + 6048 n s + 70000 n r s + 8272 n s - 10000 r s - 2284 s - 10854 n 6 6 5 5 2 4 2 - 12630 n r + 63348 n s + 75780 n r s - 152154 n s - 189450 n r s 4 3 3 3 3 4 2 4 + 190440 n s + 252600 n r s - 127290 n s - 189450 n r s 2 5 5 6 6 7 6 + 38484 n s + 75780 n r s - 198 n s - 12630 r s - 1776 s + 9645 n 5 5 4 4 2 3 2 + 10220 n r - 47650 n s - 51100 n r s + 93575 n s + 102200 n r s 3 3 2 3 2 4 4 5 - 90700 n s - 102200 n r s + 42475 n s + 51100 n r s - 6770 n s 5 6 5 4 4 3 - 10220 r s - 575 s - 5220 n - 5220 n r + 20880 n s + 20880 n r s 3 2 2 2 2 3 3 4 - 31320 n s - 31320 n r s + 20880 n s + 20880 n r s - 5220 n s 4 4 3 3 2 2 2 - 5220 r s + 1584 n + 1584 n r - 4752 n s - 4752 n r s + 4752 n s 2 3 3 3 2 2 + 4752 n r s - 1584 n s - 1584 r s - 216 n - 216 n r + 432 n s 2 2 / 6 + 432 n r s - 216 n s - 216 r s ) A[n - 4](r + 1, s - 1) / (n / 4 4 3 4 13 12 (n - 1 - s) (n - 2 - s) (n - 3 - s) ) - s (15 n - 165 n s 11 2 10 3 9 4 8 5 7 6 + 825 n s - 2475 n s + 4950 n s - 6930 n s + 6930 n s 6 7 5 8 4 9 3 10 2 11 12 - 4950 n s + 2475 n s - 825 n s + 165 n s - 15 n s - 300 n 11 10 2 9 3 8 4 7 5 + 3060 n s - 14100 n s + 38700 n s - 70200 n s + 88200 n s 6 6 5 7 4 8 3 9 2 10 - 78120 n s + 48600 n s - 20700 n s + 5700 n s - 900 n s 11 11 10 9 2 8 3 + 60 n s + 2550 n - 23970 n s + 101045 n s - 251505 n s 7 4 6 5 5 6 4 7 3 8 + 409320 n s - 455280 n s + 350910 n s - 185670 n s + 65130 n s 2 9 10 11 10 9 - 14070 n s + 1605 n s - 65 s - 12000 n + 103200 n s 8 2 7 3 6 4 5 5 - 394500 n s + 880800 n s - 1268400 n s + 1226400 n s 4 6 3 7 2 8 9 10 - 802200 n s + 348000 n s - 94800 n s + 14400 n s - 900 s 9 8 7 2 6 3 5 4 + 34095 n - 265941 n s + 911958 n s - 1801338 n s + 2253636 n s 4 5 3 6 2 7 8 9 - 1846740 n s + 987546 n s - 330678 n s + 62493 n s - 5031 s 8 7 6 2 5 3 4 4 - 59700 n + 417900 n s - 1268180 n s + 2176380 n s - 2306700 n s 3 5 2 6 7 8 7 + 1543300 n s - 635100 n s + 146580 n s - 14480 s + 62700 n 6 5 2 4 3 3 4 - 388740 n s + 1025760 n s - 1492200 n s + 1291500 n s 2 5 6 7 6 5 - 664500 n s + 188040 n s - 22560 s - 36000 n + 194400 n s 4 2 3 3 2 4 5 6 - 435600 n s + 518400 n s - 345600 n s + 122400 n s - 18000 s 5 4 3 2 2 3 4 5 + 8640 n - 39744 n s + 73152 n s - 67392 n s + 31104 n s - 5760 s ) / 6 2 3 3 3 A[n - 4](r, s) / (n (n - 4 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) ) / 12 11 11 10 10 2 9 2 + x (5 n + 15 n r - 45 n s - 165 n r s + 165 n s + 825 n r s 9 3 8 3 7 4 7 5 6 5 - 275 n s - 2475 n r s + 4950 n r s + 990 n s - 6930 n r s 6 6 5 6 5 7 4 7 4 8 - 2310 n s + 6930 n r s + 2970 n s - 4950 n r s - 2475 n s 3 8 3 9 2 9 2 10 10 + 2475 n r s + 1375 n s - 825 n r s - 495 n s + 165 n r s 11 11 12 11 10 10 + 105 n s - 15 r s - 10 s - 90 n - 240 n r + 750 n s 9 9 2 8 2 8 3 7 3 + 2400 n r s - 2550 n s - 10800 n r s + 4050 n s + 28800 n r s 7 4 6 4 6 5 5 5 5 6 - 900 n s - 50400 n r s - 8820 n s + 60480 n r s + 18900 n s 4 6 4 7 3 7 3 8 - 50400 n r s - 20700 n s + 28800 n r s + 13950 n s 2 8 2 9 9 10 10 - 10800 n r s - 5850 n s + 2400 n r s + 1410 n s - 240 r s 11 10 9 9 8 8 2 - 150 s + 680 n + 1595 n r - 5205 n s - 14355 n r s + 16245 n s 7 2 7 3 6 3 6 4 + 57420 n r s - 24180 n s - 133980 n r s + 8820 n s 5 4 5 5 4 5 4 6 + 200970 n r s + 29610 n s - 200970 n r s - 58170 n s 3 6 3 7 2 7 2 8 + 133980 n r s + 52380 n s - 57420 n r s - 26820 n s 8 9 9 10 9 8 + 14355 n r s + 7555 n s - 1595 r s - 915 s - 2800 n - 5700 n r 8 7 7 2 6 2 6 3 + 19500 n s + 45600 n r s - 55200 n s - 159600 n r s + 75600 n s 5 3 5 4 4 4 4 5 + 319200 n r s - 33600 n s - 399000 n r s - 46200 n s 3 5 3 6 2 6 2 7 + 319200 n r s + 84000 n s - 159600 n r s - 58800 n s 7 8 8 9 8 7 + 45600 n r s + 20400 n s - 5700 r s - 2900 s + 6819 n + 11850 n r 7 6 6 2 5 2 5 3 - 42702 n s - 82950 n r s + 107982 n s + 248850 n r s - 133014 n s 4 3 4 4 3 4 3 5 - 414750 n r s + 62580 n s + 414750 n r s + 32886 n s 2 5 2 6 6 7 7 - 248850 n r s - 57918 n s + 82950 n r s + 28398 n s - 11850 r s 8 7 6 6 5 - 5031 s - 9950 n - 14480 n r + 55170 n s + 86880 n r s 5 2 4 2 4 3 3 3 - 122070 n s - 217200 n r s + 131050 n s + 289600 n r s 3 4 2 4 2 5 5 6 - 58650 n s - 217200 n r s - 8250 n s + 86880 n r s + 17230 n s 6 7 6 5 5 4 - 14480 r s - 4530 s + 8360 n + 10020 n r - 40140 n s - 50100 n r s 4 2 3 2 3 3 2 3 + 75300 n s + 100200 n r s - 67000 n s - 100200 n r s 2 4 4 5 5 6 5 + 25200 n s + 50100 n r s - 60 n s - 10020 r s - 1660 s - 3600 n 4 4 3 3 2 2 2 - 3600 n r + 14400 n s + 14400 n r s - 21600 n s - 21600 n r s 2 3 3 4 4 4 3 + 14400 n s + 14400 n r s - 3600 n s - 3600 r s + 576 n + 576 n r 3 2 2 2 2 3 3 - 1728 n s - 1728 n r s + 1728 n s + 1728 n r s - 576 n s - 576 r s / 6 2 3 3 ) A[n - 5](r + 1, s - 1) / (n (n - 4 - s) (n - 1 - s) (n - 2 - s) / 3 5 10 9 8 2 7 3 6 4 (n - 3 - s) ) + s (6 n - 54 n s + 216 n s - 504 n s + 756 n s 5 5 4 6 3 7 2 8 9 9 8 - 756 n s + 504 n s - 216 n s + 54 n s - 6 n s - 90 n + 735 n s 7 2 6 3 5 4 4 5 3 6 - 2640 n s + 5460 n s - 7140 n s + 6090 n s - 3360 n s 2 7 8 9 8 7 6 2 + 1140 n s - 210 n s + 15 s + 510 n - 3740 n s + 11900 n s 5 3 4 4 3 5 2 6 7 - 21420 n s + 23800 n s - 16660 n s + 7140 n s - 1700 n s 8 7 6 5 2 4 3 3 4 + 170 s - 1350 n + 8775 n s - 24300 n s + 37125 n s - 33750 n s 2 5 6 7 6 5 4 2 + 18225 n s - 5400 n s + 675 s + 1644 n - 9316 n s + 21920 n s 3 3 2 4 5 6 5 4 - 27400 n s + 19180 n s - 7124 n s + 1096 s - 720 n + 3480 n s 3 2 2 3 4 5 / 6 - 6720 n s + 6480 n s - 3120 n s + 600 s ) A[n - 5](r, s) / (n / 2 2 2 2 10 (n - 4 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 5 - s)) - x (n 9 9 8 8 2 7 2 7 3 + 6 n r - 4 n s - 54 n r s - 9 n s + 216 n r s + 96 n s 6 3 6 4 5 4 5 5 4 5 - 504 n r s - 294 n s + 756 n r s + 504 n s - 756 n r s 4 6 3 6 3 7 2 7 2 8 - 546 n s + 504 n r s + 384 n s - 216 n r s - 171 n s 8 9 9 10 9 8 8 + 54 n r s + 44 n s - 6 r s - 5 s - 15 n - 75 n r + 60 n s 7 7 2 6 2 6 3 5 3 + 600 n r s + 60 n s - 2100 n r s - 840 n s + 4200 n r s 5 4 4 4 4 5 3 5 3 6 + 2310 n s - 5250 n r s - 3360 n s + 4200 n r s + 2940 n s 2 6 2 7 7 8 8 9 - 2100 n r s - 1560 n s + 600 n r s + 465 n s - 75 r s - 60 s 8 7 7 6 5 2 5 3 + 85 n + 340 n r - 340 n s - 2380 n r s + 7140 n r s + 2380 n s 4 3 4 4 3 4 3 5 2 5 - 11900 n r s - 5950 n s + 11900 n r s + 7140 n s - 7140 n r s 2 6 6 7 7 8 7 - 4760 n s + 2380 n r s + 1700 n s - 340 r s - 255 s - 225 n 6 6 5 5 2 4 2 - 675 n r + 900 n s + 4050 n r s - 675 n s - 10125 n r s 4 3 3 3 3 4 2 4 2 5 - 2250 n s + 13500 n r s + 5625 n s - 10125 n r s - 5400 n s 5 6 6 7 6 5 + 4050 n r s + 2475 n s - 675 r s - 450 s + 274 n + 548 n r 5 4 4 2 3 2 2 3 - 1096 n s - 2740 n r s + 1370 n s + 5480 n r s - 5480 n r s 2 4 4 5 5 6 5 - 1370 n s + 2740 n r s + 1096 n s - 548 r s - 274 s - 120 n 4 4 3 3 2 2 2 2 3 - 120 n r + 480 n s + 480 n r s - 720 n s - 720 n r s + 480 n s 3 4 4 / 6 + 480 n r s - 120 n s - 120 r s ) A[n - 6](r + 1, s - 1) / (n / 2 2 2 2 (n - 4 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 5 - s)) 6 5 4 3 2 2 3 4 5 s (n - 5 n s + 10 n s - 10 n s + 5 n s - s ) A[n - 6](r, s) - ------------------------------------------------------------------- + x 6 n (n - 4 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 5 - s) 5 5 4 4 2 3 2 3 3 2 3 (n r + n s - 5 n r s - 5 n s + 10 n r s + 10 n s - 10 n r s 2 4 4 5 5 6 / 6 - 10 n s + 5 n r s + 5 n s - r s - s ) A[n - 7](r + 1, s - 1) / (n / (n - 4 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 5 - s)) and in Maple notation A[n](r,s) = x*(n+r)/n^6*A[n-1](r+1,s-1)+s*(6*n^10-30*n^9*s+60*n^8*s^2-60*n^7*s^ 3+30*n^6*s^4-6*n^5*s^5-30*n^9+135*n^8*s-240*n^7*s^2+210*n^6*s^3-90*n^5*s^4+15*n ^4*s^5+60*n^8-240*n^7*s+380*n^6*s^2-300*n^5*s^3+120*n^4*s^4-20*n^3*s^5-60*n^7+ 210*n^6*s-300*n^5*s^2+225*n^4*s^3-90*n^3*s^4+15*n^2*s^5+30*n^6-90*n^5*s+120*n^4 *s^2-90*n^3*s^3+36*n^2*s^4-6*n*s^5-6*n^5+15*n^4*s-20*n^3*s^2+15*n^2*s^3-6*n*s^4 +s^5)/n^6/(n-1-s)^5*A[n-1](r,s)-x*(5*n^6+6*n^5*r-24*n^5*s-30*n^4*r*s+45*n^4*s^2 +60*n^3*r*s^2-40*n^3*s^3-60*n^2*r*s^3+15*n^2*s^4+30*n*r*s^4-6*r*s^5-s^6-15*n^5-\ 15*n^4*r+60*n^4*s+60*n^3*r*s-90*n^3*s^2-90*n^2*r*s^2+60*n^2*s^3+60*n*r*s^3-15*n *s^4-15*r*s^4+20*n^4+20*n^3*r-60*n^3*s-60*n^2*r*s+60*n^2*s^2+60*n*r*s^2-20*n*s^ 3-20*r*s^3-15*n^3-15*n^2*r+30*n^2*s+30*n*r*s-15*n*s^2-15*r*s^2+6*n^2+6*n*r-6*n* s-6*r*s-n-r)/n^6/(n-1-s)^5*A[n-2](r+1,s-1)-s^2*(15*n^13-135*n^12*s+540*n^11*s^2 -1260*n^10*s^3+1890*n^9*s^4-1890*n^8*s^5+1260*n^7*s^6-540*n^6*s^7+135*n^5*s^8-\ 15*n^4*s^9-180*n^12+1500*n^11*s-5520*n^10*s^2+11760*n^9*s^3-15960*n^8*s^4+14280 *n^7*s^5-8400*n^6*s^6+3120*n^5*s^7-660*n^4*s^8+60*n^3*s^9+930*n^11-7130*n^10*s+ 23975*n^9*s^2-46305*n^8*s^3+56455*n^7*s^4-44905*n^6*s^5+23205*n^5*s^6-7475*n^4* s^7+1355*n^3*s^8-105*n^2*s^9-2700*n^10+18900*n^9*s-57600*n^8*s^2+99990*n^7*s^3-\ 108540*n^6*s^4+76050*n^5*s^5-34200*n^4*s^6+9450*n^3*s^7-1440*n^2*s^8+90*n*s^9+ 4815*n^9-30495*n^8*s+83415*n^7*s^2-128781*n^6*s^3+122986*n^5*s^4-74780*n^4*s^5+ 28615*n^3*s^6-6505*n^2*s^7+761*n*s^8-31*s^9-5400*n^8+30600*n^7*s-74250*n^6*s^2+ 100620*n^5*s^3-83130*n^4*s^4+42720*n^3*s^5-13230*n^2*s^6+2220*n*s^7-150*s^8+ 3720*n^7-18600*n^6*s+39480*n^5*s^2-46200*n^4*s^3+32200*n^3*s^4-13320*n^2*s^5+ 3000*n*s^6-280*s^7-1440*n^6+6240*n^5*s-11400*n^4*s^2+11280*n^3*s^3-6360*n^2*s^4 +1920*n*s^5-240*s^6+240*n^5-880*n^4*s+1360*n^3*s^2-1104*n^2*s^3+464*n*s^4-80*s^ 5)/n^6/(n-1-s)^5/(n-2-s)^4*A[n-2](r,s)+x*(10*n^10+15*n^9*r-85*n^9*s-135*n^8*r*s +315*n^8*s^2+540*n^7*r*s^2-660*n^7*s^3-1260*n^6*r*s^3+840*n^6*s^4+1890*n^5*r*s^ 4-630*n^5*s^5-1890*n^4*r*s^5+210*n^4*s^6+1260*n^3*r*s^6+60*n^3*s^7-540*n^2*r*s^ 7-90*n^2*s^8+135*n*r*s^8+35*n*s^9-15*r*s^9-5*s^10-90*n^9-120*n^8*r+690*n^8*s+ 960*n^7*r*s-2280*n^7*s^2-3360*n^6*r*s^2+4200*n^6*s^3+6720*n^5*r*s^3-4620*n^5*s^ 4-8400*n^4*r*s^4+2940*n^4*s^5+6720*n^3*r*s^5-840*n^3*s^6-3360*n^2*r*s^6-120*n^2 *s^7+960*n*r*s^7+150*n*s^8-120*r*s^8-30*s^9+345*n^8+415*n^7*r-2345*n^7*s-2905*n ^6*r*s+6755*n^6*s^2+8715*n^5*r*s^2-10605*n^5*s^3-14525*n^4*r*s^3+9625*n^4*s^4+ 14525*n^3*r*s^4-4795*n^3*s^5-8715*n^2*r*s^5+945*n^2*s^6+2905*n*r*s^6+145*n*s^7-\ 415*r*s^7-70*s^8-735*n^7-810*n^6*r+4335*n^6*s+4860*n^5*r*s-10575*n^5*s^2-12150* n^4*r*s^2+13575*n^4*s^3+16200*n^3*r*s^3-9525*n^3*s^4-12150*n^2*r*s^4+3285*n^2*s ^5+4860*n*r*s^5-285*n*s^6-810*r*s^6-75*s^7+954*n^6+985*n^5*r-4739*n^5*s-4925*n^ 4*r*s+9385*n^4*s^2+9850*n^3*r*s^2-9230*n^3*s^3-9850*n^2*r*s^3+4460*n^2*s^4+4925 *n*r*s^4-799*n*s^5-985*r*s^5-31*s^6-780*n^5-780*n^4*r+3120*n^4*s+3120*n^3*r*s-\ 4680*n^3*s^2-4680*n^2*r*s^2+3120*n^2*s^3+3120*n*r*s^3-780*n*s^4-780*r*s^4+400*n ^4+400*n^3*r-1200*n^3*s-1200*n^2*r*s+1200*n^2*s^2+1200*n*r*s^2-400*n*s^3-400*r* s^3-120*n^3-120*n^2*r+240*n^2*s+240*n*r*s-120*n*s^2-120*r*s^2+16*n^2+16*n*r-16* n*s-16*r*s)/n^6/(n-1-s)^5/(n-2-s)^4*A[n-3](r+1,s-1)+2*s^3*(10*n^14-110*n^13*s+ 550*n^12*s^2-1650*n^11*s^3+3300*n^10*s^4-4620*n^9*s^5+4620*n^8*s^6-3300*n^7*s^7 +1650*n^6*s^8-550*n^5*s^9+110*n^4*s^10-10*n^3*s^11-180*n^13+1845*n^12*s-8550*n^ 11*s^2+23625*n^10*s^3-43200*n^9*s^4+54810*n^8*s^5-49140*n^7*s^6+31050*n^6*s^7-\ 13500*n^5*s^8+3825*n^4*s^9-630*n^3*s^10+45*n^2*s^11+1410*n^12-13395*n^11*s+ 57180*n^10*s^2-144495*n^9*s^3+239580*n^8*s^4-272790*n^7*s^5+216720*n^6*s^6-\ 119430*n^5*s^7+44370*n^4*s^8-10455*n^3*s^9+1380*n^2*s^10-75*n*s^11-6300*n^11+ 55125*n^10*s-215190*n^9*s^2+493065*n^8*s^3-733680*n^7*s^4+740250*n^6*s^5-512820 *n^5*s^6+241290*n^4*s^7-74340*n^3*s^8+13905*n^2*s^9-1350*n*s^10+45*s^11+17670*n ^10-141360*n^9*s+500238*n^8*s^2-1028344*n^7*s^3+1355284*n^6*s^4-1191288*n^5*s^5 +703220*n^4*s^6-273128*n^3*s^7+66054*n^2*s^8-8824*n*s^9+478*s^10-32220*n^9+ 233595*n^8*s-741402*n^7*s^2+1349319*n^6*s^3-1548072*n^5*s^4+1157715*n^4*s^5-\ 562290*n^3*s^6+170217*n^2*s^7-28944*n*s^8+2082*s^9+38150*n^8-247975*n^7*s+ 696560*n^6*s^2-1103175*n^5*s^3+1076100*n^4*s^4-661125*n^3*s^5+249400*n^2*s^6-\ 52685*n*s^7+4750*s^8-28260*n^7+162495*n^6*s-397170*n^5*s^2+534915*n^4*s^3-\ 428760*n^3*s^4+204525*n^2*s^5-53730*n*s^6+5985*s^7+11880*n^6-59400*n^5*s+123552 *n^4*s^2-137016*n^3*s^3+85536*n^2*s^4-28512*n*s^5+3960*s^6-2160*n^5+9180*n^4*s-\ 15768*n^3*s^2+13716*n^2*s^3-6048*n*s^4+1080*s^5)/n^6/(n-1-s)^4/(n-2-s)^4/(n-3-s )^3*A[n-3](r,s)-x*(10*n^12+20*n^11*r-100*n^11*s-220*n^10*r*s+440*n^10*s^2+1100* n^9*r*s^2-1100*n^9*s^3-3300*n^8*r*s^3+1650*n^8*s^4+6600*n^7*r*s^4-1320*n^7*s^5-\ 9240*n^6*r*s^5+9240*n^5*r*s^6+1320*n^5*s^7-6600*n^4*r*s^7-1650*n^4*s^8+3300*n^3 *r*s^8+1100*n^3*s^9-1100*n^2*r*s^9-440*n^2*s^10+220*n*r*s^10+100*n*s^11-20*r*s^ 11-10*s^12-150*n^11-270*n^10*r+1380*n^10*s+2700*n^9*r*s-5550*n^9*s^2-12150*n^8* r*s^2+12600*n^8*s^3+32400*n^7*r*s^3-17100*n^7*s^4-56700*n^6*r*s^4+12600*n^6*s^5 +68040*n^5*r*s^5-1260*n^5*s^6-56700*n^4*r*s^6-7200*n^4*s^7+32400*n^3*r*s^7+7650 *n^3*s^8-12150*n^2*r*s^8-3900*n^2*s^9+2700*n*r*s^9+1050*n*s^10-270*r*s^10-120*s ^11+965*n^10+1560*n^9*r-8090*n^9*s-14040*n^8*r*s+29385*n^8*s^2+56160*n^7*r*s^2-\ 59640*n^7*s^3-131040*n^6*r*s^3+71610*n^6*s^4+196560*n^5*r*s^4-46620*n^5*s^5-\ 196560*n^4*r*s^5+6090*n^4*s^6+131040*n^3*r*s^6+15240*n^3*s^7-56160*n^2*r*s^7-\ 12735*n^2*s^8+14040*n*r*s^8+4390*n*s^9-1560*r*s^9-595*s^10-3480*n^9-5040*n^8*r+ 26280*n^8*s+40320*n^7*r*s-84960*n^7*s^2-141120*n^6*r*s^2+151200*n^6*s^3+282240* n^5*r*s^3-156240*n^5*s^4-352800*n^4*r*s^4+85680*n^4*s^5+282240*n^3*r*s^5-10080* n^3*s^6-141120*n^2*r*s^6-15840*n^2*s^7+40320*n*r*s^7+9000*n*s^8-5040*r*s^8-1560 *s^9+7716*n^8+10000*n^7*r-51728*n^7*s-70000*n^6*r*s+146048*n^6*s^2+210000*n^5*r *s^2-222096*n^5*s^3-350000*n^4*r*s^3+190120*n^4*s^4+350000*n^3*r*s^4-82096*n^3* s^5-210000*n^2*r*s^5+6048*n^2*s^6+70000*n*r*s^6+8272*n*s^7-10000*r*s^7-2284*s^8 -10854*n^7-12630*n^6*r+63348*n^6*s+75780*n^5*r*s-152154*n^5*s^2-189450*n^4*r*s^ 2+190440*n^4*s^3+252600*n^3*r*s^3-127290*n^3*s^4-189450*n^2*r*s^4+38484*n^2*s^5 +75780*n*r*s^5-198*n*s^6-12630*r*s^6-1776*s^7+9645*n^6+10220*n^5*r-47650*n^5*s-\ 51100*n^4*r*s+93575*n^4*s^2+102200*n^3*r*s^2-90700*n^3*s^3-102200*n^2*r*s^3+ 42475*n^2*s^4+51100*n*r*s^4-6770*n*s^5-10220*r*s^5-575*s^6-5220*n^5-5220*n^4*r+ 20880*n^4*s+20880*n^3*r*s-31320*n^3*s^2-31320*n^2*r*s^2+20880*n^2*s^3+20880*n*r *s^3-5220*n*s^4-5220*r*s^4+1584*n^4+1584*n^3*r-4752*n^3*s-4752*n^2*r*s+4752*n^2 *s^2+4752*n*r*s^2-1584*n*s^3-1584*r*s^3-216*n^3-216*n^2*r+432*n^2*s+432*n*r*s-\ 216*n*s^2-216*r*s^2)/n^6/(n-1-s)^4/(n-2-s)^4/(n-3-s)^3*A[n-4](r+1,s-1)-s^4*(15* n^13-165*n^12*s+825*n^11*s^2-2475*n^10*s^3+4950*n^9*s^4-6930*n^8*s^5+6930*n^7*s ^6-4950*n^6*s^7+2475*n^5*s^8-825*n^4*s^9+165*n^3*s^10-15*n^2*s^11-300*n^12+3060 *n^11*s-14100*n^10*s^2+38700*n^9*s^3-70200*n^8*s^4+88200*n^7*s^5-78120*n^6*s^6+ 48600*n^5*s^7-20700*n^4*s^8+5700*n^3*s^9-900*n^2*s^10+60*n*s^11+2550*n^11-23970 *n^10*s+101045*n^9*s^2-251505*n^8*s^3+409320*n^7*s^4-455280*n^6*s^5+350910*n^5* s^6-185670*n^4*s^7+65130*n^3*s^8-14070*n^2*s^9+1605*n*s^10-65*s^11-12000*n^10+ 103200*n^9*s-394500*n^8*s^2+880800*n^7*s^3-1268400*n^6*s^4+1226400*n^5*s^5-\ 802200*n^4*s^6+348000*n^3*s^7-94800*n^2*s^8+14400*n*s^9-900*s^10+34095*n^9-\ 265941*n^8*s+911958*n^7*s^2-1801338*n^6*s^3+2253636*n^5*s^4-1846740*n^4*s^5+ 987546*n^3*s^6-330678*n^2*s^7+62493*n*s^8-5031*s^9-59700*n^8+417900*n^7*s-\ 1268180*n^6*s^2+2176380*n^5*s^3-2306700*n^4*s^4+1543300*n^3*s^5-635100*n^2*s^6+ 146580*n*s^7-14480*s^8+62700*n^7-388740*n^6*s+1025760*n^5*s^2-1492200*n^4*s^3+ 1291500*n^3*s^4-664500*n^2*s^5+188040*n*s^6-22560*s^7-36000*n^6+194400*n^5*s-\ 435600*n^4*s^2+518400*n^3*s^3-345600*n^2*s^4+122400*n*s^5-18000*s^6+8640*n^5-\ 39744*n^4*s+73152*n^3*s^2-67392*n^2*s^3+31104*n*s^4-5760*s^5)/n^6/(n-4-s)^2/(n-\ 1-s)^3/(n-2-s)^3/(n-3-s)^3*A[n-4](r,s)+x*(5*n^12+15*n^11*r-45*n^11*s-165*n^10*r *s+165*n^10*s^2+825*n^9*r*s^2-275*n^9*s^3-2475*n^8*r*s^3+4950*n^7*r*s^4+990*n^7 *s^5-6930*n^6*r*s^5-2310*n^6*s^6+6930*n^5*r*s^6+2970*n^5*s^7-4950*n^4*r*s^7-\ 2475*n^4*s^8+2475*n^3*r*s^8+1375*n^3*s^9-825*n^2*r*s^9-495*n^2*s^10+165*n*r*s^ 10+105*n*s^11-15*r*s^11-10*s^12-90*n^11-240*n^10*r+750*n^10*s+2400*n^9*r*s-2550 *n^9*s^2-10800*n^8*r*s^2+4050*n^8*s^3+28800*n^7*r*s^3-900*n^7*s^4-50400*n^6*r*s ^4-8820*n^6*s^5+60480*n^5*r*s^5+18900*n^5*s^6-50400*n^4*r*s^6-20700*n^4*s^7+ 28800*n^3*r*s^7+13950*n^3*s^8-10800*n^2*r*s^8-5850*n^2*s^9+2400*n*r*s^9+1410*n* s^10-240*r*s^10-150*s^11+680*n^10+1595*n^9*r-5205*n^9*s-14355*n^8*r*s+16245*n^8 *s^2+57420*n^7*r*s^2-24180*n^7*s^3-133980*n^6*r*s^3+8820*n^6*s^4+200970*n^5*r*s ^4+29610*n^5*s^5-200970*n^4*r*s^5-58170*n^4*s^6+133980*n^3*r*s^6+52380*n^3*s^7-\ 57420*n^2*r*s^7-26820*n^2*s^8+14355*n*r*s^8+7555*n*s^9-1595*r*s^9-915*s^10-2800 *n^9-5700*n^8*r+19500*n^8*s+45600*n^7*r*s-55200*n^7*s^2-159600*n^6*r*s^2+75600* n^6*s^3+319200*n^5*r*s^3-33600*n^5*s^4-399000*n^4*r*s^4-46200*n^4*s^5+319200*n^ 3*r*s^5+84000*n^3*s^6-159600*n^2*r*s^6-58800*n^2*s^7+45600*n*r*s^7+20400*n*s^8-\ 5700*r*s^8-2900*s^9+6819*n^8+11850*n^7*r-42702*n^7*s-82950*n^6*r*s+107982*n^6*s ^2+248850*n^5*r*s^2-133014*n^5*s^3-414750*n^4*r*s^3+62580*n^4*s^4+414750*n^3*r* s^4+32886*n^3*s^5-248850*n^2*r*s^5-57918*n^2*s^6+82950*n*r*s^6+28398*n*s^7-\ 11850*r*s^7-5031*s^8-9950*n^7-14480*n^6*r+55170*n^6*s+86880*n^5*r*s-122070*n^5* s^2-217200*n^4*r*s^2+131050*n^4*s^3+289600*n^3*r*s^3-58650*n^3*s^4-217200*n^2*r *s^4-8250*n^2*s^5+86880*n*r*s^5+17230*n*s^6-14480*r*s^6-4530*s^7+8360*n^6+10020 *n^5*r-40140*n^5*s-50100*n^4*r*s+75300*n^4*s^2+100200*n^3*r*s^2-67000*n^3*s^3-\ 100200*n^2*r*s^3+25200*n^2*s^4+50100*n*r*s^4-60*n*s^5-10020*r*s^5-1660*s^6-3600 *n^5-3600*n^4*r+14400*n^4*s+14400*n^3*r*s-21600*n^3*s^2-21600*n^2*r*s^2+14400*n ^2*s^3+14400*n*r*s^3-3600*n*s^4-3600*r*s^4+576*n^4+576*n^3*r-1728*n^3*s-1728*n^ 2*r*s+1728*n^2*s^2+1728*n*r*s^2-576*n*s^3-576*r*s^3)/n^6/(n-4-s)^2/(n-1-s)^3/(n -2-s)^3/(n-3-s)^3*A[n-5](r+1,s-1)+s^5*(6*n^10-54*n^9*s+216*n^8*s^2-504*n^7*s^3+ 756*n^6*s^4-756*n^5*s^5+504*n^4*s^6-216*n^3*s^7+54*n^2*s^8-6*n*s^9-90*n^9+735*n ^8*s-2640*n^7*s^2+5460*n^6*s^3-7140*n^5*s^4+6090*n^4*s^5-3360*n^3*s^6+1140*n^2* s^7-210*n*s^8+15*s^9+510*n^8-3740*n^7*s+11900*n^6*s^2-21420*n^5*s^3+23800*n^4*s ^4-16660*n^3*s^5+7140*n^2*s^6-1700*n*s^7+170*s^8-1350*n^7+8775*n^6*s-24300*n^5* s^2+37125*n^4*s^3-33750*n^3*s^4+18225*n^2*s^5-5400*n*s^6+675*s^7+1644*n^6-9316* n^5*s+21920*n^4*s^2-27400*n^3*s^3+19180*n^2*s^4-7124*n*s^5+1096*s^6-720*n^5+ 3480*n^4*s-6720*n^3*s^2+6480*n^2*s^3-3120*n*s^4+600*s^5)/n^6/(n-4-s)^2/(n-1-s)^ 2/(n-2-s)^2/(n-3-s)^2/(n-5-s)*A[n-5](r,s)-x*(n^10+6*n^9*r-4*n^9*s-54*n^8*r*s-9* n^8*s^2+216*n^7*r*s^2+96*n^7*s^3-504*n^6*r*s^3-294*n^6*s^4+756*n^5*r*s^4+504*n^ 5*s^5-756*n^4*r*s^5-546*n^4*s^6+504*n^3*r*s^6+384*n^3*s^7-216*n^2*r*s^7-171*n^2 *s^8+54*n*r*s^8+44*n*s^9-6*r*s^9-5*s^10-15*n^9-75*n^8*r+60*n^8*s+600*n^7*r*s+60 *n^7*s^2-2100*n^6*r*s^2-840*n^6*s^3+4200*n^5*r*s^3+2310*n^5*s^4-5250*n^4*r*s^4-\ 3360*n^4*s^5+4200*n^3*r*s^5+2940*n^3*s^6-2100*n^2*r*s^6-1560*n^2*s^7+600*n*r*s^ 7+465*n*s^8-75*r*s^8-60*s^9+85*n^8+340*n^7*r-340*n^7*s-2380*n^6*r*s+7140*n^5*r* s^2+2380*n^5*s^3-11900*n^4*r*s^3-5950*n^4*s^4+11900*n^3*r*s^4+7140*n^3*s^5-7140 *n^2*r*s^5-4760*n^2*s^6+2380*n*r*s^6+1700*n*s^7-340*r*s^7-255*s^8-225*n^7-675*n ^6*r+900*n^6*s+4050*n^5*r*s-675*n^5*s^2-10125*n^4*r*s^2-2250*n^4*s^3+13500*n^3* r*s^3+5625*n^3*s^4-10125*n^2*r*s^4-5400*n^2*s^5+4050*n*r*s^5+2475*n*s^6-675*r*s ^6-450*s^7+274*n^6+548*n^5*r-1096*n^5*s-2740*n^4*r*s+1370*n^4*s^2+5480*n^3*r*s^ 2-5480*n^2*r*s^3-1370*n^2*s^4+2740*n*r*s^4+1096*n*s^5-548*r*s^5-274*s^6-120*n^5 -120*n^4*r+480*n^4*s+480*n^3*r*s-720*n^3*s^2-720*n^2*r*s^2+480*n^2*s^3+480*n*r* s^3-120*n*s^4-120*r*s^4)/n^6/(n-4-s)^2/(n-1-s)^2/(n-2-s)^2/(n-3-s)^2/(n-5-s)*A[ n-6](r+1,s-1)-s^6*(n^5-5*n^4*s+10*n^3*s^2-10*n^2*s^3+5*n*s^4-s^5)/n^6/(n-4-s)/( n-1-s)/(n-2-s)/(n-3-s)/(n-5-s)*A[n-6](r,s)+x*(n^5*r+n^5*s-5*n^4*r*s-5*n^4*s^2+ 10*n^3*r*s^2+10*n^3*s^3-10*n^2*r*s^3-10*n^2*s^4+5*n*r*s^4+5*n*s^5-r*s^5-s^6)/n^ 6/(n-4-s)/(n-1-s)/(n-2-s)/(n-3-s)/(n-5-s)*A[n-7](r+1,s-1) Proof: We claim that: Let a(n,k,r,s) be the summand on the sum defining A[n](r,s) in other words k (k - 1 + p) (n - k + q) x (r + k) (s - k) a(n, k, r, s) = ---------------------------------------- 6 (k!) (n - k)! and in Maple notation a(n,k,r,s) = 1/k!^6/(n-k)!*x^k*(r+k)^(k-1+p)*(s-k)^(n-k+q) Then the following identity is true 5 (n - s + 6) (s + r) a(n, k, r, s) ----------------------------------------------------------------------- - (n - s + 5) (n - s + 4) (n - s + 3) (n - s + 2) (n + r + 6) (n - s + 1) 4 6 5 4 4 2 3 2 3 3 (n - s + 6) (n + 6 n r - 30 n r s - 15 n s + 60 n r s + 40 n s 2 3 2 4 4 5 5 6 5 - 60 n r s - 45 n s + 30 n r s + 24 n s - 6 r s - 5 s + 21 n 4 3 3 2 2 2 2 3 3 + 105 n r - 420 n r s - 210 n s + 630 n r s + 420 n s - 420 n r s 4 4 5 4 3 2 - 315 n s + 105 r s + 84 s + 175 n + 700 n r - 2100 n r s 2 2 2 3 3 4 3 - 1050 n s + 2100 n r s + 1400 n s - 700 r s - 525 s + 735 n 2 2 2 3 2 + 2205 n r - 4410 n r s - 2205 n s + 2205 r s + 1470 s + 1624 n 2 + 3248 n r - 3248 r s - 1624 s + 1764 n + 1764 r + 720) a(n + 1, k, r, s) / 2 2 2 / ((n - s + 1) (n - s + 5) (n + r + 6) (n - s + 4) (n - s + 3) / 2 (n - s + 2) ) - 6 5 (s + 1) (n - s + 6) a(n + 1, k + 1, r - 1, s + 1) ------------------------------------------------------------------------- x (n - s + 5) (n - s + 4) (n - s + 3) (n - s + 2) (n + r + 6) (n - s + 1) 3 9 8 8 7 7 2 + (n - s + 6) (5 n + 15 n r - 30 n s - 120 n r s + 60 n s 6 2 5 3 5 4 4 4 4 5 + 420 n r s - 840 n r s - 210 n s + 1050 n r s + 420 n s 3 5 3 6 2 6 2 7 7 8 - 840 n r s - 420 n s + 420 n r s + 240 n s - 120 n r s - 75 n s 8 9 8 7 7 6 + 15 r s + 10 s + 180 n + 480 n r - 960 n s - 3360 n r s 6 2 5 2 4 3 4 4 3 4 + 1680 n s + 10080 n r s - 16800 n r s - 4200 n s + 16800 n r s 3 5 2 5 2 6 6 7 + 6720 n s - 10080 n r s - 5040 n s + 3360 n r s + 1920 n s 7 8 7 6 6 5 - 480 r s - 300 s + 2840 n + 6635 n r - 13245 n s - 39810 n r s 5 2 4 2 4 3 3 3 3 4 + 19830 n s + 99525 n r s + 125 n s - 132700 n r s - 33300 n s 2 4 2 5 5 6 6 + 99525 n r s + 39885 n s - 39810 n r s - 19930 n s + 6635 r s 7 6 5 5 4 + 3795 s + 25760 n + 51720 n r - 102840 n s - 258600 n r s 4 2 3 2 3 3 2 3 + 127800 n s + 517200 n r s + 2000 n s - 517200 n r s 2 4 4 5 5 6 - 130800 n s + 258600 n r s + 104040 n s - 51720 r s - 25960 s 5 4 4 3 3 2 + 147939 n + 248550 n r - 491145 n s - 994200 n r s + 485190 n s 2 2 2 3 3 4 + 1491300 n r s + 11910 n s - 994200 n r s - 254505 n s 4 5 4 3 3 + 248550 r s + 100611 s + 557500 n + 753760 n r - 1476240 n s 2 2 2 2 3 - 2261280 n r s + 1083720 n s + 2261280 n r s + 31280 n s 3 4 3 2 2 - 753760 r s - 196260 s + 1377680 n + 1408260 n r - 2724780 n s 2 2 3 2 - 2816520 n r s + 1316520 n s + 1408260 r s + 30580 s + 2151360 n 2 + 1481760 n r - 2820960 n s - 1481760 r s + 669600 s + 1925136 n / 2 + 672336 r - 1252800 s + 751680) a(2 + n, k, r, s) / ((n - s + 2) / 3 3 3 5 6 (n - s + 5) (n + r + 6) (n - s + 4) (n - s + 3) ) + (s + 1) (6 n 5 4 2 3 3 2 4 5 5 4 - 30 n s + 60 n s - 60 n s + 30 n s - 6 n s + 132 n - 555 n s 3 2 2 3 4 5 4 3 + 900 n s - 690 n s + 240 n s - 27 s + 1155 n - 3920 n s 2 2 3 4 3 2 2 + 4830 n s - 2520 n s + 455 s + 5110 n - 13125 n s + 10920 n s 3 2 2 - 2905 s + 11949 n - 20650 n s + 8701 s + 13832 n - 12068 s + 6084) 4 / 2 (n - s + 6) a(2 + n, k + 1, r - 1, s + 1) / (x (n - s + 1) (n - s + 5) / 2 2 2 2 10 (n - s + 4) (n - s + 3) (n + r + 6) (n - s + 2) ) - (n - s + 6) (10 n 9 9 8 8 2 7 2 7 3 + 20 n r - 80 n s - 180 n r s + 270 n s + 720 n r s - 480 n s 6 3 6 4 5 4 4 5 4 6 - 1680 n r s + 420 n s + 2520 n r s - 2520 n r s - 420 n s 3 6 3 7 2 7 2 8 8 + 1680 n r s + 480 n s - 720 n r s - 270 n s + 180 n r s 9 9 10 9 8 8 7 + 80 n s - 20 r s - 10 s + 450 n + 810 n r - 3240 n s - 6480 n r s 7 2 6 2 6 3 5 3 5 4 + 9720 n s + 22680 n r s - 15120 n s - 45360 n r s + 11340 n s 4 4 3 5 3 6 2 6 2 7 + 56700 n r s - 45360 n r s - 7560 n s + 22680 n r s + 6480 n s 7 8 8 9 8 7 - 6480 n r s - 2430 n s + 810 r s + 360 s + 9065 n + 14520 n r 7 6 6 2 5 2 - 58000 n s - 101640 n r s + 152180 n s + 304920 n r s 5 3 4 3 4 4 3 4 - 202720 n s - 508200 n r s + 126350 n s + 508200 n r s 3 5 2 5 2 6 6 7 + 560 n s - 304920 n r s - 51100 n s + 101640 n r s + 29120 n s 7 8 7 6 6 - 14520 r s - 5455 s + 107640 n + 151200 n r - 602280 n s 5 5 2 4 2 4 3 - 907200 n r s + 1353240 n s + 2268000 n r s - 1499400 n s 3 3 3 4 2 4 2 5 - 3024000 n r s + 743400 n s + 2268000 n r s + 7560 n s 5 6 6 7 6 - 907200 n r s - 153720 n s + 151200 r s + 43560 s + 834276 n 5 5 4 4 2 + 1007920 n r - 3997736 n s - 5039600 n r s + 7474540 n s 3 2 3 3 2 3 2 4 + 10079200 n r s - 6606320 n s - 10079200 n r s + 2434940 n s 4 5 5 6 5 + 5039600 n r s + 33944 n s - 1007920 r s - 173644 s + 4409802 n 4 4 3 3 2 + 4460490 n r - 17588520 n s - 17841960 n r s + 26256060 n s 2 2 2 3 3 4 + 26762940 n r s - 17335080 n s - 17841960 n r s + 4207050 n s 4 5 4 3 3 + 4460490 r s + 50688 s + 16097865 n + 13104860 n r - 51286600 n s 2 2 2 2 3 - 39314580 n r s + 57272610 n s + 39314580 n r s - 25076880 n s 3 4 3 2 2 - 13104860 r s + 2993005 s + 40071780 n + 24649380 n r - 95565960 n s 2 2 3 - 49298760 n r s + 70916580 n s + 24649380 r s - 15422400 s 2 2 + 65093184 n + 26936784 n r - 103249584 n s - 26936784 r s + 38156400 s / + 62305848 n + 13031928 r - 49273920 s + 26684640) a(n + 3, k, r, s) / ( / 3 4 4 4 10 (n - s + 3) (n - s + 4) (n - s + 5) (n + r + 6)) - (s + 1) (15 n 9 8 2 7 3 6 4 5 5 4 6 - 120 n s + 420 n s - 840 n s + 1050 n s - 840 n s + 420 n s 3 7 2 8 9 8 7 2 6 3 - 120 n s + 15 n s + 630 n - 4560 n s + 14280 n s - 25200 n s 5 4 4 5 3 6 2 7 8 + 27300 n s - 18480 n s + 7560 n s - 1680 n s + 150 n s 8 7 6 2 5 3 4 4 + 11745 n - 75960 n s + 209495 n s - 319530 n s + 291675 n s 3 5 2 6 7 8 7 - 160420 n s + 50385 n s - 7770 n s + 380 s + 127920 n 6 5 2 4 3 3 4 - 727610 n s + 1731060 n s - 2218350 n s + 1637800 n s 2 5 6 7 6 5 - 686070 n s + 147060 n s - 11810 s + 900890 n - 4414266 n s 4 2 3 3 2 4 5 6 + 8806530 n s - 9101260 n s + 5093910 n s - 1444170 n s + 158366 s 5 4 3 2 2 3 + 4284354 n - 17579670 n s + 28228900 n s - 22052220 n s 4 5 4 3 2 2 + 8314650 n s - 1196014 s + 13926150 n - 45931000 n s + 55644360 n s 3 4 3 2 2 - 29200320 n s + 5560810 s + 30533440 n - 75875640 n s + 61632720 n s 3 2 2 - 16290520 s + 43193460 n - 71872584 n s + 29351460 s + 35582976 n 3 / - 29728224 s + 12958416) (n - s + 6) a(n + 3, k + 1, r - 1, s + 1) / (x / 2 3 3 3 (n - s + 2) (n - s + 5) (n - s + 4) (n + r + 6) (n - s + 3) ) + 9 8 8 7 7 2 (n - s + 6) (10 n + 15 n r - 75 n s - 120 n r s + 240 n s 6 2 6 3 5 3 5 4 4 4 + 420 n r s - 420 n s - 840 n r s + 420 n s + 1050 n r s 4 5 3 5 2 6 2 7 7 8 - 210 n s - 840 n r s + 420 n r s + 60 n s - 120 n r s - 30 n s 8 9 8 7 7 6 + 15 r s + 5 s + 450 n + 600 n r - 3000 n s - 4200 n r s 6 2 5 2 5 3 4 3 4 4 + 8400 n s + 12600 n r s - 12600 n s - 21000 n r s + 10500 n s 3 4 3 5 2 5 6 7 + 21000 n r s - 4200 n s - 12600 n r s + 4200 n r s + 600 n s 7 8 7 6 6 5 - 600 r s - 150 s + 8985 n + 10495 n r - 52400 n s - 62970 n r s 5 2 4 2 4 3 3 3 + 125715 n s + 157425 n r s - 157050 n s - 209900 n r s 3 4 2 4 2 5 5 6 + 104575 n s + 157425 n r s - 31260 n s - 62970 n r s - 75 n s 6 7 6 5 5 + 10495 r s + 1510 s + 104475 n + 104850 n r - 522000 n s 4 4 2 3 2 3 3 - 524250 n r s + 1042875 n s + 1048500 n r s - 1041000 n s 2 3 2 4 4 5 - 1048500 n r s + 518625 n s + 524250 n r s - 102600 n s 5 6 5 4 4 - 104850 r s - 375 s + 779634 n + 654385 n r - 3243785 n s 3 3 2 2 2 2 3 - 2617540 n r s + 5178800 n s + 3926310 n r s - 3870030 n s 3 4 4 5 4 - 2617540 n r s + 1280630 n s + 654385 r s - 125249 s + 3872100 n 3 3 2 2 2 + 2612700 n r - 12875700 n s - 7838100 n r s + 15394500 n s 2 3 3 4 3 + 7838100 n r s - 7650300 n s - 2612700 r s + 1259400 s + 12799120 n 2 2 2 + 6517120 n r - 31880240 n s - 13034240 n r s + 25363120 n s 2 3 2 + 6517120 r s - 6282000 s + 27151800 n + 9286200 n r - 45017400 n s 2 - 9286200 r s + 17865600 s + 33543376 n + 5787376 r - 27756000 s / 5 4 + 18386880) a(4 + n, k, r, s) / ((n - s + 5) (n + r + 6) (n - s + 4) ) / 3 12 11 10 2 9 3 8 4 + 2 (s + 1) (10 n - 90 n s + 360 n s - 840 n s + 1260 n s 7 5 6 6 5 7 4 8 3 9 11 - 1260 n s + 840 n s - 360 n s + 90 n s - 10 n s + 570 n 10 9 2 8 3 7 4 6 5 - 4725 n s + 17280 n s - 36540 n s + 49140 n s - 43470 n s 5 6 4 7 3 8 2 9 10 + 25200 n s - 9180 n s + 1890 n s - 165 n s + 14805 n 9 8 2 7 3 6 4 5 5 - 112095 n s + 371040 n s - 702240 n s + 833490 n s - 639030 n s 4 6 3 7 2 8 9 9 + 313320 n s - 93240 n s + 14865 n s - 915 n s + 231685 n 8 7 2 6 3 5 4 - 1586085 n s + 4692720 n s - 7824600 n s + 8028930 n s 4 5 3 6 2 7 8 9 - 5187210 n s + 2065560 n s - 471120 n s + 51825 n s - 1705 s 8 7 6 2 5 3 + 2432745 n - 14871000 n s + 38709768 n s - 55697368 n s 4 4 3 5 2 6 7 + 48033170 n s - 25103080 n s + 7611520 n s - 1183208 n s 8 7 6 5 2 + 67453 s + 18055380 n - 97001949 n s + 217590000 n s 4 3 3 4 2 5 6 - 262628820 n s + 182715960 n s - 72408705 n s + 14858844 n s 7 6 5 4 2 - 1180710 s + 97115633 n - 449147181 n s + 843985740 n s 3 3 2 4 5 6 - 820231380 n s + 431517015 n s - 115240743 n s + 12000916 s 5 4 3 2 2 3 + 381416111 n - 1476169105 n s + 2230387940 n s - 1635985610 n s 4 5 4 3 + 578403265 n s - 78052601 s + 1085515245 n - 3374615130 n s 2 2 3 4 3 + 3843007542 n s - 1890762282 n s + 336854625 s + 2183203330 n 2 2 3 2 - 5110268556 n s + 3898249842 n s - 964668652 s + 2945295702 n 2 - 4613409540 n s + 1767715650 s + 2393023536 n - 1880943804 s + 885553884 2 / 3 ) (n - s + 6) a(4 + n, k + 1, r - 1, s + 1) / (x (n - s + 3) / 4 4 6 5 5 (n + r + 6) (n - s + 5) (n - s + 4) ) - (5 n + 6 n r - 24 n s 4 4 2 3 2 3 3 2 3 2 4 - 30 n r s + 45 n s + 60 n r s - 40 n s - 60 n r s + 15 n s 4 5 6 5 4 4 3 + 30 n r s - 6 r s - s + 165 n + 165 n r - 660 n s - 660 n r s 3 2 2 2 2 3 3 4 4 + 990 n s + 990 n r s - 660 n s - 660 n r s + 165 n s + 165 r s 4 3 3 2 2 2 2 + 2270 n + 1820 n r - 7260 n s - 5460 n r s + 8160 n s + 5460 n r s 3 3 4 3 2 2 - 3620 n s - 1820 r s + 450 s + 16665 n + 10065 n r - 39930 n s 2 2 3 2 - 20130 n r s + 29865 n s + 10065 r s - 6600 s + 68856 n + 27906 n r 2 - 109806 n s - 27906 r s + 40950 s + 151811 n + 31031 r - 120780 s / 5 2 + 139530) a(n + 5, k, r, s) / ((n - s + 5) (n + r + 6)) - (s + 1) / 12 11 10 2 9 3 8 4 (n - s + 6) (15 n - 120 n s + 420 n s - 840 n s + 1050 n s 7 5 6 6 5 7 4 8 11 10 - 840 n s + 420 n s - 120 n s + 15 n s + 960 n - 7080 n s 9 2 8 3 7 4 6 5 5 6 + 22680 n s - 41160 n s + 46200 n s - 32760 n s + 14280 n s 4 7 3 8 10 9 8 2 - 3480 n s + 360 n s + 28080 n - 189320 n s + 549495 n s 7 3 6 4 5 5 4 6 3 7 - 893730 n s + 886825 n s - 546180 n s + 201945 n s - 40370 n s 2 8 9 8 7 2 6 3 + 3255 n s + 496360 n - 3028530 n s + 7865610 n s - 11285840 n s 5 4 4 5 3 6 2 7 + 9698400 n s - 5045010 n s + 1519930 n s - 234060 n s 8 8 7 6 2 + 13140 n s + 5905455 n - 32202270 n s + 73661865 n s 5 3 4 4 3 5 2 6 - 91331796 n s + 66082990 n s - 27877330 n s + 6418965 n s 7 8 7 6 5 2 - 677860 n s + 19981 s + 49819350 n - 238969130 n s + 471576792 n s 4 3 3 4 2 5 6 - 491174120 n s + 287246610 n s - 92132430 n s + 14416880 n s 7 6 5 4 2 - 783952 s + 305576380 n - 1262892888 n s + 2089984650 n s 3 3 2 4 5 6 - 1755291800 n s + 777767800 n s - 168592640 n s + 13448498 s 5 4 3 2 2 3 + 1373092584 n - 4752891980 n s + 6331563440 n s - 4019285000 n s 4 5 4 3 + 1199270880 n s - 131749924 s + 4486031425 n - 12483661220 n s 2 2 3 4 3 + 12548040870 n s - 5350818724 n s + 806195025 s + 10392588740 n 2 2 3 2 - 21793731260 n s + 14690000668 n s - 3155387364 s + 16205347360 n 2 - 22760092352 n s + 7714449088 s + 15271736856 n - 10772151512 s / 4 + 6577964656) a(n + 5, k + 1, r - 1, s + 1) / (x (n - s + 4) / 5 (n - s + 5) (n + r + 6)) + a(n + 6, k, r, s) + (s + 1) 2 4 3 2 2 3 (2 n - 2 n s + 24 n - 13 s + 71) (3 n - 6 n s + 3 n s + 72 n 2 2 2 2 - 111 n s + 39 n s + 645 n - 681 n s + 127 s + 2556 n - 1384 s + 3781) 4 3 2 2 3 2 2 2 2 (n - 2 n s + n s + 24 n - 37 n s + 13 n s + 215 n - 227 n s + 43 s / 5 + 852 n - 460 s + 1261) a(n + 6, k + 1, r - 1, s + 1) / (x (n - s + 5) / 6 (n + 7) a(n + 7, k + 1, r - 1, s + 1) (n + r + 6)) - -------------------------------------- = 0 x (n + r + 6) and in Maple notation: (n-s+6)^5*(s+r)/(n-s+5)/(n-s+4)/(n-s+3)/(n-s+2)/(n+r+6)/(n-s+1)*a(n,k,r,s)-(n-s +6)^4*(n^6+6*n^5*r-30*n^4*r*s-15*n^4*s^2+60*n^3*r*s^2+40*n^3*s^3-60*n^2*r*s^3-\ 45*n^2*s^4+30*n*r*s^4+24*n*s^5-6*r*s^5-5*s^6+21*n^5+105*n^4*r-420*n^3*r*s-210*n ^3*s^2+630*n^2*r*s^2+420*n^2*s^3-420*n*r*s^3-315*n*s^4+105*r*s^4+84*s^5+175*n^4 +700*n^3*r-2100*n^2*r*s-1050*n^2*s^2+2100*n*r*s^2+1400*n*s^3-700*r*s^3-525*s^4+ 735*n^3+2205*n^2*r-4410*n*r*s-2205*n*s^2+2205*r*s^2+1470*s^3+1624*n^2+3248*n*r-\ 3248*r*s-1624*s^2+1764*n+1764*r+720)/(n-s+1)/(n-s+5)^2/(n+r+6)/(n-s+4)^2/(n-s+3 )^2/(n-s+2)^2*a(n+1,k,r,s)-(s+1)^6*(n-s+6)^5/x/(n-s+5)/(n-s+4)/(n-s+3)/(n-s+2)/ (n+r+6)/(n-s+1)*a(n+1,k+1,r-1,s+1)+(n-s+6)^3*(5*n^9+15*n^8*r-30*n^8*s-120*n^7*r *s+60*n^7*s^2+420*n^6*r*s^2-840*n^5*r*s^3-210*n^5*s^4+1050*n^4*r*s^4+420*n^4*s^ 5-840*n^3*r*s^5-420*n^3*s^6+420*n^2*r*s^6+240*n^2*s^7-120*n*r*s^7-75*n*s^8+15*r *s^8+10*s^9+180*n^8+480*n^7*r-960*n^7*s-3360*n^6*r*s+1680*n^6*s^2+10080*n^5*r*s ^2-16800*n^4*r*s^3-4200*n^4*s^4+16800*n^3*r*s^4+6720*n^3*s^5-10080*n^2*r*s^5-\ 5040*n^2*s^6+3360*n*r*s^6+1920*n*s^7-480*r*s^7-300*s^8+2840*n^7+6635*n^6*r-\ 13245*n^6*s-39810*n^5*r*s+19830*n^5*s^2+99525*n^4*r*s^2+125*n^4*s^3-132700*n^3* r*s^3-33300*n^3*s^4+99525*n^2*r*s^4+39885*n^2*s^5-39810*n*r*s^5-19930*n*s^6+ 6635*r*s^6+3795*s^7+25760*n^6+51720*n^5*r-102840*n^5*s-258600*n^4*r*s+127800*n^ 4*s^2+517200*n^3*r*s^2+2000*n^3*s^3-517200*n^2*r*s^3-130800*n^2*s^4+258600*n*r* s^4+104040*n*s^5-51720*r*s^5-25960*s^6+147939*n^5+248550*n^4*r-491145*n^4*s-\ 994200*n^3*r*s+485190*n^3*s^2+1491300*n^2*r*s^2+11910*n^2*s^3-994200*n*r*s^3-\ 254505*n*s^4+248550*r*s^4+100611*s^5+557500*n^4+753760*n^3*r-1476240*n^3*s-\ 2261280*n^2*r*s+1083720*n^2*s^2+2261280*n*r*s^2+31280*n*s^3-753760*r*s^3-196260 *s^4+1377680*n^3+1408260*n^2*r-2724780*n^2*s-2816520*n*r*s+1316520*n*s^2+ 1408260*r*s^2+30580*s^3+2151360*n^2+1481760*n*r-2820960*n*s-1481760*r*s+669600* s^2+1925136*n+672336*r-1252800*s+751680)/(n-s+2)^2/(n-s+5)^3/(n+r+6)/(n-s+4)^3/ (n-s+3)^3*a(2+n,k,r,s)+(s+1)^5*(6*n^6-30*n^5*s+60*n^4*s^2-60*n^3*s^3+30*n^2*s^4 -6*n*s^5+132*n^5-555*n^4*s+900*n^3*s^2-690*n^2*s^3+240*n*s^4-27*s^5+1155*n^4-\ 3920*n^3*s+4830*n^2*s^2-2520*n*s^3+455*s^4+5110*n^3-13125*n^2*s+10920*n*s^2-\ 2905*s^3+11949*n^2-20650*n*s+8701*s^2+13832*n-12068*s+6084)*(n-s+6)^4/x/(n-s+1) /(n-s+5)^2/(n-s+4)^2/(n-s+3)^2/(n+r+6)/(n-s+2)^2*a(2+n,k+1,r-1,s+1)-(n-s+6)^2*( 10*n^10+20*n^9*r-80*n^9*s-180*n^8*r*s+270*n^8*s^2+720*n^7*r*s^2-480*n^7*s^3-\ 1680*n^6*r*s^3+420*n^6*s^4+2520*n^5*r*s^4-2520*n^4*r*s^5-420*n^4*s^6+1680*n^3*r *s^6+480*n^3*s^7-720*n^2*r*s^7-270*n^2*s^8+180*n*r*s^8+80*n*s^9-20*r*s^9-10*s^ 10+450*n^9+810*n^8*r-3240*n^8*s-6480*n^7*r*s+9720*n^7*s^2+22680*n^6*r*s^2-15120 *n^6*s^3-45360*n^5*r*s^3+11340*n^5*s^4+56700*n^4*r*s^4-45360*n^3*r*s^5-7560*n^3 *s^6+22680*n^2*r*s^6+6480*n^2*s^7-6480*n*r*s^7-2430*n*s^8+810*r*s^8+360*s^9+ 9065*n^8+14520*n^7*r-58000*n^7*s-101640*n^6*r*s+152180*n^6*s^2+304920*n^5*r*s^2 -202720*n^5*s^3-508200*n^4*r*s^3+126350*n^4*s^4+508200*n^3*r*s^4+560*n^3*s^5-\ 304920*n^2*r*s^5-51100*n^2*s^6+101640*n*r*s^6+29120*n*s^7-14520*r*s^7-5455*s^8+ 107640*n^7+151200*n^6*r-602280*n^6*s-907200*n^5*r*s+1353240*n^5*s^2+2268000*n^4 *r*s^2-1499400*n^4*s^3-3024000*n^3*r*s^3+743400*n^3*s^4+2268000*n^2*r*s^4+7560* n^2*s^5-907200*n*r*s^5-153720*n*s^6+151200*r*s^6+43560*s^7+834276*n^6+1007920*n ^5*r-3997736*n^5*s-5039600*n^4*r*s+7474540*n^4*s^2+10079200*n^3*r*s^2-6606320*n ^3*s^3-10079200*n^2*r*s^3+2434940*n^2*s^4+5039600*n*r*s^4+33944*n*s^5-1007920*r *s^5-173644*s^6+4409802*n^5+4460490*n^4*r-17588520*n^4*s-17841960*n^3*r*s+ 26256060*n^3*s^2+26762940*n^2*r*s^2-17335080*n^2*s^3-17841960*n*r*s^3+4207050*n *s^4+4460490*r*s^4+50688*s^5+16097865*n^4+13104860*n^3*r-51286600*n^3*s-\ 39314580*n^2*r*s+57272610*n^2*s^2+39314580*n*r*s^2-25076880*n*s^3-13104860*r*s^ 3+2993005*s^4+40071780*n^3+24649380*n^2*r-95565960*n^2*s-49298760*n*r*s+ 70916580*n*s^2+24649380*r*s^2-15422400*s^3+65093184*n^2+26936784*n*r-103249584* n*s-26936784*r*s+38156400*s^2+62305848*n+13031928*r-49273920*s+26684640)/(n-s+3 )^3/(n-s+4)^4/(n-s+5)^4/(n+r+6)*a(n+3,k,r,s)-(s+1)^4*(15*n^10-120*n^9*s+420*n^8 *s^2-840*n^7*s^3+1050*n^6*s^4-840*n^5*s^5+420*n^4*s^6-120*n^3*s^7+15*n^2*s^8+ 630*n^9-4560*n^8*s+14280*n^7*s^2-25200*n^6*s^3+27300*n^5*s^4-18480*n^4*s^5+7560 *n^3*s^6-1680*n^2*s^7+150*n*s^8+11745*n^8-75960*n^7*s+209495*n^6*s^2-319530*n^5 *s^3+291675*n^4*s^4-160420*n^3*s^5+50385*n^2*s^6-7770*n*s^7+380*s^8+127920*n^7-\ 727610*n^6*s+1731060*n^5*s^2-2218350*n^4*s^3+1637800*n^3*s^4-686070*n^2*s^5+ 147060*n*s^6-11810*s^7+900890*n^6-4414266*n^5*s+8806530*n^4*s^2-9101260*n^3*s^3 +5093910*n^2*s^4-1444170*n*s^5+158366*s^6+4284354*n^5-17579670*n^4*s+28228900*n ^3*s^2-22052220*n^2*s^3+8314650*n*s^4-1196014*s^5+13926150*n^4-45931000*n^3*s+ 55644360*n^2*s^2-29200320*n*s^3+5560810*s^4+30533440*n^3-75875640*n^2*s+ 61632720*n*s^2-16290520*s^3+43193460*n^2-71872584*n*s+29351460*s^2+35582976*n-\ 29728224*s+12958416)*(n-s+6)^3/x/(n-s+2)^2/(n-s+5)^3/(n-s+4)^3/(n+r+6)/(n-s+3)^ 3*a(n+3,k+1,r-1,s+1)+(n-s+6)*(10*n^9+15*n^8*r-75*n^8*s-120*n^7*r*s+240*n^7*s^2+ 420*n^6*r*s^2-420*n^6*s^3-840*n^5*r*s^3+420*n^5*s^4+1050*n^4*r*s^4-210*n^4*s^5-\ 840*n^3*r*s^5+420*n^2*r*s^6+60*n^2*s^7-120*n*r*s^7-30*n*s^8+15*r*s^8+5*s^9+450* n^8+600*n^7*r-3000*n^7*s-4200*n^6*r*s+8400*n^6*s^2+12600*n^5*r*s^2-12600*n^5*s^ 3-21000*n^4*r*s^3+10500*n^4*s^4+21000*n^3*r*s^4-4200*n^3*s^5-12600*n^2*r*s^5+ 4200*n*r*s^6+600*n*s^7-600*r*s^7-150*s^8+8985*n^7+10495*n^6*r-52400*n^6*s-62970 *n^5*r*s+125715*n^5*s^2+157425*n^4*r*s^2-157050*n^4*s^3-209900*n^3*r*s^3+104575 *n^3*s^4+157425*n^2*r*s^4-31260*n^2*s^5-62970*n*r*s^5-75*n*s^6+10495*r*s^6+1510 *s^7+104475*n^6+104850*n^5*r-522000*n^5*s-524250*n^4*r*s+1042875*n^4*s^2+ 1048500*n^3*r*s^2-1041000*n^3*s^3-1048500*n^2*r*s^3+518625*n^2*s^4+524250*n*r*s ^4-102600*n*s^5-104850*r*s^5-375*s^6+779634*n^5+654385*n^4*r-3243785*n^4*s-\ 2617540*n^3*r*s+5178800*n^3*s^2+3926310*n^2*r*s^2-3870030*n^2*s^3-2617540*n*r*s ^3+1280630*n*s^4+654385*r*s^4-125249*s^5+3872100*n^4+2612700*n^3*r-12875700*n^3 *s-7838100*n^2*r*s+15394500*n^2*s^2+7838100*n*r*s^2-7650300*n*s^3-2612700*r*s^3 +1259400*s^4+12799120*n^3+6517120*n^2*r-31880240*n^2*s-13034240*n*r*s+25363120* n*s^2+6517120*r*s^2-6282000*s^3+27151800*n^2+9286200*n*r-45017400*n*s-9286200*r *s+17865600*s^2+33543376*n+5787376*r-27756000*s+18386880)/(n-s+5)^5/(n+r+6)/(n- s+4)^4*a(4+n,k,r,s)+2*(s+1)^3*(10*n^12-90*n^11*s+360*n^10*s^2-840*n^9*s^3+1260* n^8*s^4-1260*n^7*s^5+840*n^6*s^6-360*n^5*s^7+90*n^4*s^8-10*n^3*s^9+570*n^11-\ 4725*n^10*s+17280*n^9*s^2-36540*n^8*s^3+49140*n^7*s^4-43470*n^6*s^5+25200*n^5*s ^6-9180*n^4*s^7+1890*n^3*s^8-165*n^2*s^9+14805*n^10-112095*n^9*s+371040*n^8*s^2 -702240*n^7*s^3+833490*n^6*s^4-639030*n^5*s^5+313320*n^4*s^6-93240*n^3*s^7+ 14865*n^2*s^8-915*n*s^9+231685*n^9-1586085*n^8*s+4692720*n^7*s^2-7824600*n^6*s^ 3+8028930*n^5*s^4-5187210*n^4*s^5+2065560*n^3*s^6-471120*n^2*s^7+51825*n*s^8-\ 1705*s^9+2432745*n^8-14871000*n^7*s+38709768*n^6*s^2-55697368*n^5*s^3+48033170* n^4*s^4-25103080*n^3*s^5+7611520*n^2*s^6-1183208*n*s^7+67453*s^8+18055380*n^7-\ 97001949*n^6*s+217590000*n^5*s^2-262628820*n^4*s^3+182715960*n^3*s^4-72408705*n ^2*s^5+14858844*n*s^6-1180710*s^7+97115633*n^6-449147181*n^5*s+843985740*n^4*s^ 2-820231380*n^3*s^3+431517015*n^2*s^4-115240743*n*s^5+12000916*s^6+381416111*n^ 5-1476169105*n^4*s+2230387940*n^3*s^2-1635985610*n^2*s^3+578403265*n*s^4-\ 78052601*s^5+1085515245*n^4-3374615130*n^3*s+3843007542*n^2*s^2-1890762282*n*s^ 3+336854625*s^4+2183203330*n^3-5110268556*n^2*s+3898249842*n*s^2-964668652*s^3+ 2945295702*n^2-4613409540*n*s+1767715650*s^2+2393023536*n-1880943804*s+ 885553884)*(n-s+6)^2/x/(n-s+3)^3/(n+r+6)/(n-s+5)^4/(n-s+4)^4*a(4+n,k+1,r-1,s+1) -(5*n^6+6*n^5*r-24*n^5*s-30*n^4*r*s+45*n^4*s^2+60*n^3*r*s^2-40*n^3*s^3-60*n^2*r *s^3+15*n^2*s^4+30*n*r*s^4-6*r*s^5-s^6+165*n^5+165*n^4*r-660*n^4*s-660*n^3*r*s+ 990*n^3*s^2+990*n^2*r*s^2-660*n^2*s^3-660*n*r*s^3+165*n*s^4+165*r*s^4+2270*n^4+ 1820*n^3*r-7260*n^3*s-5460*n^2*r*s+8160*n^2*s^2+5460*n*r*s^2-3620*n*s^3-1820*r* s^3+450*s^4+16665*n^3+10065*n^2*r-39930*n^2*s-20130*n*r*s+29865*n*s^2+10065*r*s ^2-6600*s^3+68856*n^2+27906*n*r-109806*n*s-27906*r*s+40950*s^2+151811*n+31031*r -120780*s+139530)/(n-s+5)^5/(n+r+6)*a(n+5,k,r,s)-(s+1)^2*(n-s+6)*(15*n^12-120*n ^11*s+420*n^10*s^2-840*n^9*s^3+1050*n^8*s^4-840*n^7*s^5+420*n^6*s^6-120*n^5*s^7 +15*n^4*s^8+960*n^11-7080*n^10*s+22680*n^9*s^2-41160*n^8*s^3+46200*n^7*s^4-\ 32760*n^6*s^5+14280*n^5*s^6-3480*n^4*s^7+360*n^3*s^8+28080*n^10-189320*n^9*s+ 549495*n^8*s^2-893730*n^7*s^3+886825*n^6*s^4-546180*n^5*s^5+201945*n^4*s^6-\ 40370*n^3*s^7+3255*n^2*s^8+496360*n^9-3028530*n^8*s+7865610*n^7*s^2-11285840*n^ 6*s^3+9698400*n^5*s^4-5045010*n^4*s^5+1519930*n^3*s^6-234060*n^2*s^7+13140*n*s^ 8+5905455*n^8-32202270*n^7*s+73661865*n^6*s^2-91331796*n^5*s^3+66082990*n^4*s^4 -27877330*n^3*s^5+6418965*n^2*s^6-677860*n*s^7+19981*s^8+49819350*n^7-238969130 *n^6*s+471576792*n^5*s^2-491174120*n^4*s^3+287246610*n^3*s^4-92132430*n^2*s^5+ 14416880*n*s^6-783952*s^7+305576380*n^6-1262892888*n^5*s+2089984650*n^4*s^2-\ 1755291800*n^3*s^3+777767800*n^2*s^4-168592640*n*s^5+13448498*s^6+1373092584*n^ 5-4752891980*n^4*s+6331563440*n^3*s^2-4019285000*n^2*s^3+1199270880*n*s^4-\ 131749924*s^5+4486031425*n^4-12483661220*n^3*s+12548040870*n^2*s^2-5350818724*n *s^3+806195025*s^4+10392588740*n^3-21793731260*n^2*s+14690000668*n*s^2-\ 3155387364*s^3+16205347360*n^2-22760092352*n*s+7714449088*s^2+15271736856*n-\ 10772151512*s+6577964656)/x/(n-s+4)^4/(n-s+5)^5/(n+r+6)*a(n+5,k+1,r-1,s+1)+a(n+ 6,k,r,s)+(s+1)*(2*n^2-2*n*s+24*n-13*s+71)*(3*n^4-6*n^3*s+3*n^2*s^2+72*n^3-111*n ^2*s+39*n*s^2+645*n^2-681*n*s+127*s^2+2556*n-1384*s+3781)*(n^4-2*n^3*s+n^2*s^2+ 24*n^3-37*n^2*s+13*n*s^2+215*n^2-227*n*s+43*s^2+852*n-460*s+1261)/x/(n-s+5)^5/( n+r+6)*a(n+6,k+1,r-1,s+1)-(n+7)^6/x/(n+r+6)*a(n+7,k+1,r-1,s+1) = 0 The proof of this identity is routine (divide by a(n,k,r,s), simplify each t\ erm,and now each term is a rational function. Now add them all up and \ verify that they add up to zero.) Now sum it from k=0 to k=n, which is the same as from k=-infinity to k=infin\ ity (since it vanishes for k<0 and k>n 5 (n - s + 6) (s + r) A[n](r, s) ----------------------------------------------------------------------- - (n - s + 5) (n - s + 4) (n - s + 3) (n - s + 2) (n + r + 6) (n - s + 1) 4 6 5 4 4 2 3 2 3 3 (n - s + 6) (n + 6 n r - 30 n r s - 15 n s + 60 n r s + 40 n s 2 3 2 4 4 5 5 6 5 - 60 n r s - 45 n s + 30 n r s + 24 n s - 6 r s - 5 s + 21 n 4 3 3 2 2 2 2 3 3 + 105 n r - 420 n r s - 210 n s + 630 n r s + 420 n s - 420 n r s 4 4 5 4 3 2 - 315 n s + 105 r s + 84 s + 175 n + 700 n r - 2100 n r s 2 2 2 3 3 4 3 - 1050 n s + 2100 n r s + 1400 n s - 700 r s - 525 s + 735 n 2 2 2 3 2 + 2205 n r - 4410 n r s - 2205 n s + 2205 r s + 1470 s + 1624 n 2 + 3248 n r - 3248 r s - 1624 s + 1764 n + 1764 r + 720) A[n + 1](r, s) / 2 2 2 / ((n - s + 1) (n - s + 5) (n + r + 6) (n - s + 4) (n - s + 3) / 2 (n - s + 2) ) - 6 5 (s + 1) (n - s + 6) A[n + 1](r - 1, s + 1) ------------------------------------------------------------------------- x (n - s + 5) (n - s + 4) (n - s + 3) (n - s + 2) (n + r + 6) (n - s + 1) 3 9 8 8 7 7 2 + (n - s + 6) (5 n + 15 n r - 30 n s - 120 n r s + 60 n s 6 2 5 3 5 4 4 4 4 5 + 420 n r s - 840 n r s - 210 n s + 1050 n r s + 420 n s 3 5 3 6 2 6 2 7 7 8 - 840 n r s - 420 n s + 420 n r s + 240 n s - 120 n r s - 75 n s 8 9 8 7 7 6 + 15 r s + 10 s + 180 n + 480 n r - 960 n s - 3360 n r s 6 2 5 2 4 3 4 4 3 4 + 1680 n s + 10080 n r s - 16800 n r s - 4200 n s + 16800 n r s 3 5 2 5 2 6 6 7 + 6720 n s - 10080 n r s - 5040 n s + 3360 n r s + 1920 n s 7 8 7 6 6 5 - 480 r s - 300 s + 2840 n + 6635 n r - 13245 n s - 39810 n r s 5 2 4 2 4 3 3 3 3 4 + 19830 n s + 99525 n r s + 125 n s - 132700 n r s - 33300 n s 2 4 2 5 5 6 6 + 99525 n r s + 39885 n s - 39810 n r s - 19930 n s + 6635 r s 7 6 5 5 4 + 3795 s + 25760 n + 51720 n r - 102840 n s - 258600 n r s 4 2 3 2 3 3 2 3 + 127800 n s + 517200 n r s + 2000 n s - 517200 n r s 2 4 4 5 5 6 - 130800 n s + 258600 n r s + 104040 n s - 51720 r s - 25960 s 5 4 4 3 3 2 + 147939 n + 248550 n r - 491145 n s - 994200 n r s + 485190 n s 2 2 2 3 3 4 + 1491300 n r s + 11910 n s - 994200 n r s - 254505 n s 4 5 4 3 3 + 248550 r s + 100611 s + 557500 n + 753760 n r - 1476240 n s 2 2 2 2 3 - 2261280 n r s + 1083720 n s + 2261280 n r s + 31280 n s 3 4 3 2 2 - 753760 r s - 196260 s + 1377680 n + 1408260 n r - 2724780 n s 2 2 3 2 - 2816520 n r s + 1316520 n s + 1408260 r s + 30580 s + 2151360 n 2 + 1481760 n r - 2820960 n s - 1481760 r s + 669600 s + 1925136 n / 2 + 672336 r - 1252800 s + 751680) A[2 + n](r, s) / ((n - s + 2) / 3 3 3 5 6 (n - s + 5) (n + r + 6) (n - s + 4) (n - s + 3) ) + (s + 1) (6 n 5 4 2 3 3 2 4 5 5 4 - 30 n s + 60 n s - 60 n s + 30 n s - 6 n s + 132 n - 555 n s 3 2 2 3 4 5 4 3 + 900 n s - 690 n s + 240 n s - 27 s + 1155 n - 3920 n s 2 2 3 4 3 2 2 + 4830 n s - 2520 n s + 455 s + 5110 n - 13125 n s + 10920 n s 3 2 2 - 2905 s + 11949 n - 20650 n s + 8701 s + 13832 n - 12068 s + 6084) 4 / 2 (n - s + 6) A[2 + n](r - 1, s + 1) / (x (n - s + 1) (n - s + 5) / 2 2 2 2 10 (n - s + 4) (n - s + 3) (n + r + 6) (n - s + 2) ) - (n - s + 6) (10 n 9 9 8 8 2 7 2 7 3 + 20 n r - 80 n s - 180 n r s + 270 n s + 720 n r s - 480 n s 6 3 6 4 5 4 4 5 4 6 - 1680 n r s + 420 n s + 2520 n r s - 2520 n r s - 420 n s 3 6 3 7 2 7 2 8 8 + 1680 n r s + 480 n s - 720 n r s - 270 n s + 180 n r s 9 9 10 9 8 8 7 + 80 n s - 20 r s - 10 s + 450 n + 810 n r - 3240 n s - 6480 n r s 7 2 6 2 6 3 5 3 5 4 + 9720 n s + 22680 n r s - 15120 n s - 45360 n r s + 11340 n s 4 4 3 5 3 6 2 6 2 7 + 56700 n r s - 45360 n r s - 7560 n s + 22680 n r s + 6480 n s 7 8 8 9 8 7 - 6480 n r s - 2430 n s + 810 r s + 360 s + 9065 n + 14520 n r 7 6 6 2 5 2 - 58000 n s - 101640 n r s + 152180 n s + 304920 n r s 5 3 4 3 4 4 3 4 - 202720 n s - 508200 n r s + 126350 n s + 508200 n r s 3 5 2 5 2 6 6 7 + 560 n s - 304920 n r s - 51100 n s + 101640 n r s + 29120 n s 7 8 7 6 6 - 14520 r s - 5455 s + 107640 n + 151200 n r - 602280 n s 5 5 2 4 2 4 3 - 907200 n r s + 1353240 n s + 2268000 n r s - 1499400 n s 3 3 3 4 2 4 2 5 - 3024000 n r s + 743400 n s + 2268000 n r s + 7560 n s 5 6 6 7 6 - 907200 n r s - 153720 n s + 151200 r s + 43560 s + 834276 n 5 5 4 4 2 + 1007920 n r - 3997736 n s - 5039600 n r s + 7474540 n s 3 2 3 3 2 3 2 4 + 10079200 n r s - 6606320 n s - 10079200 n r s + 2434940 n s 4 5 5 6 5 + 5039600 n r s + 33944 n s - 1007920 r s - 173644 s + 4409802 n 4 4 3 3 2 + 4460490 n r - 17588520 n s - 17841960 n r s + 26256060 n s 2 2 2 3 3 4 + 26762940 n r s - 17335080 n s - 17841960 n r s + 4207050 n s 4 5 4 3 3 + 4460490 r s + 50688 s + 16097865 n + 13104860 n r - 51286600 n s 2 2 2 2 3 - 39314580 n r s + 57272610 n s + 39314580 n r s - 25076880 n s 3 4 3 2 2 - 13104860 r s + 2993005 s + 40071780 n + 24649380 n r - 95565960 n s 2 2 3 - 49298760 n r s + 70916580 n s + 24649380 r s - 15422400 s 2 2 + 65093184 n + 26936784 n r - 103249584 n s - 26936784 r s + 38156400 s / + 62305848 n + 13031928 r - 49273920 s + 26684640) A[n + 3](r, s) / ( / 3 4 4 4 10 (n - s + 3) (n - s + 4) (n - s + 5) (n + r + 6)) - (s + 1) (15 n 9 8 2 7 3 6 4 5 5 4 6 - 120 n s + 420 n s - 840 n s + 1050 n s - 840 n s + 420 n s 3 7 2 8 9 8 7 2 6 3 - 120 n s + 15 n s + 630 n - 4560 n s + 14280 n s - 25200 n s 5 4 4 5 3 6 2 7 8 + 27300 n s - 18480 n s + 7560 n s - 1680 n s + 150 n s 8 7 6 2 5 3 4 4 + 11745 n - 75960 n s + 209495 n s - 319530 n s + 291675 n s 3 5 2 6 7 8 7 - 160420 n s + 50385 n s - 7770 n s + 380 s + 127920 n 6 5 2 4 3 3 4 - 727610 n s + 1731060 n s - 2218350 n s + 1637800 n s 2 5 6 7 6 5 - 686070 n s + 147060 n s - 11810 s + 900890 n - 4414266 n s 4 2 3 3 2 4 5 6 + 8806530 n s - 9101260 n s + 5093910 n s - 1444170 n s + 158366 s 5 4 3 2 2 3 + 4284354 n - 17579670 n s + 28228900 n s - 22052220 n s 4 5 4 3 2 2 + 8314650 n s - 1196014 s + 13926150 n - 45931000 n s + 55644360 n s 3 4 3 2 2 - 29200320 n s + 5560810 s + 30533440 n - 75875640 n s + 61632720 n s 3 2 2 - 16290520 s + 43193460 n - 71872584 n s + 29351460 s + 35582976 n 3 / - 29728224 s + 12958416) (n - s + 6) A[n + 3](r - 1, s + 1) / (x / 2 3 3 3 (n - s + 2) (n - s + 5) (n - s + 4) (n + r + 6) (n - s + 3) ) + 9 8 8 7 7 2 (n - s + 6) (10 n + 15 n r - 75 n s - 120 n r s + 240 n s 6 2 6 3 5 3 5 4 4 4 + 420 n r s - 420 n s - 840 n r s + 420 n s + 1050 n r s 4 5 3 5 2 6 2 7 7 8 - 210 n s - 840 n r s + 420 n r s + 60 n s - 120 n r s - 30 n s 8 9 8 7 7 6 + 15 r s + 5 s + 450 n + 600 n r - 3000 n s - 4200 n r s 6 2 5 2 5 3 4 3 4 4 + 8400 n s + 12600 n r s - 12600 n s - 21000 n r s + 10500 n s 3 4 3 5 2 5 6 7 + 21000 n r s - 4200 n s - 12600 n r s + 4200 n r s + 600 n s 7 8 7 6 6 5 - 600 r s - 150 s + 8985 n + 10495 n r - 52400 n s - 62970 n r s 5 2 4 2 4 3 3 3 + 125715 n s + 157425 n r s - 157050 n s - 209900 n r s 3 4 2 4 2 5 5 6 + 104575 n s + 157425 n r s - 31260 n s - 62970 n r s - 75 n s 6 7 6 5 5 + 10495 r s + 1510 s + 104475 n + 104850 n r - 522000 n s 4 4 2 3 2 3 3 - 524250 n r s + 1042875 n s + 1048500 n r s - 1041000 n s 2 3 2 4 4 5 - 1048500 n r s + 518625 n s + 524250 n r s - 102600 n s 5 6 5 4 4 - 104850 r s - 375 s + 779634 n + 654385 n r - 3243785 n s 3 3 2 2 2 2 3 - 2617540 n r s + 5178800 n s + 3926310 n r s - 3870030 n s 3 4 4 5 4 - 2617540 n r s + 1280630 n s + 654385 r s - 125249 s + 3872100 n 3 3 2 2 2 + 2612700 n r - 12875700 n s - 7838100 n r s + 15394500 n s 2 3 3 4 3 + 7838100 n r s - 7650300 n s - 2612700 r s + 1259400 s + 12799120 n 2 2 2 + 6517120 n r - 31880240 n s - 13034240 n r s + 25363120 n s 2 3 2 + 6517120 r s - 6282000 s + 27151800 n + 9286200 n r - 45017400 n s 2 - 9286200 r s + 17865600 s + 33543376 n + 5787376 r - 27756000 s / 5 4 + 18386880) A[4 + n](r, s) / ((n - s + 5) (n + r + 6) (n - s + 4) ) + 2 / 3 12 11 10 2 9 3 8 4 (s + 1) (10 n - 90 n s + 360 n s - 840 n s + 1260 n s 7 5 6 6 5 7 4 8 3 9 11 - 1260 n s + 840 n s - 360 n s + 90 n s - 10 n s + 570 n 10 9 2 8 3 7 4 6 5 - 4725 n s + 17280 n s - 36540 n s + 49140 n s - 43470 n s 5 6 4 7 3 8 2 9 10 + 25200 n s - 9180 n s + 1890 n s - 165 n s + 14805 n 9 8 2 7 3 6 4 5 5 - 112095 n s + 371040 n s - 702240 n s + 833490 n s - 639030 n s 4 6 3 7 2 8 9 9 + 313320 n s - 93240 n s + 14865 n s - 915 n s + 231685 n 8 7 2 6 3 5 4 - 1586085 n s + 4692720 n s - 7824600 n s + 8028930 n s 4 5 3 6 2 7 8 9 - 5187210 n s + 2065560 n s - 471120 n s + 51825 n s - 1705 s 8 7 6 2 5 3 + 2432745 n - 14871000 n s + 38709768 n s - 55697368 n s 4 4 3 5 2 6 7 + 48033170 n s - 25103080 n s + 7611520 n s - 1183208 n s 8 7 6 5 2 + 67453 s + 18055380 n - 97001949 n s + 217590000 n s 4 3 3 4 2 5 6 - 262628820 n s + 182715960 n s - 72408705 n s + 14858844 n s 7 6 5 4 2 - 1180710 s + 97115633 n - 449147181 n s + 843985740 n s 3 3 2 4 5 6 - 820231380 n s + 431517015 n s - 115240743 n s + 12000916 s 5 4 3 2 2 3 + 381416111 n - 1476169105 n s + 2230387940 n s - 1635985610 n s 4 5 4 3 + 578403265 n s - 78052601 s + 1085515245 n - 3374615130 n s 2 2 3 4 3 + 3843007542 n s - 1890762282 n s + 336854625 s + 2183203330 n 2 2 3 2 - 5110268556 n s + 3898249842 n s - 964668652 s + 2945295702 n 2 - 4613409540 n s + 1767715650 s + 2393023536 n - 1880943804 s + 885553884 2 / 3 ) (n - s + 6) A[4 + n](r - 1, s + 1) / (x (n - s + 3) (n + r + 6) / 4 4 6 5 5 4 (n - s + 5) (n - s + 4) ) - (5 n + 6 n r - 24 n s - 30 n r s 4 2 3 2 3 3 2 3 2 4 4 + 45 n s + 60 n r s - 40 n s - 60 n r s + 15 n s + 30 n r s 5 6 5 4 4 3 3 2 - 6 r s - s + 165 n + 165 n r - 660 n s - 660 n r s + 990 n s 2 2 2 3 3 4 4 4 + 990 n r s - 660 n s - 660 n r s + 165 n s + 165 r s + 2270 n 3 3 2 2 2 2 + 1820 n r - 7260 n s - 5460 n r s + 8160 n s + 5460 n r s 3 3 4 3 2 2 - 3620 n s - 1820 r s + 450 s + 16665 n + 10065 n r - 39930 n s 2 2 3 2 - 20130 n r s + 29865 n s + 10065 r s - 6600 s + 68856 n + 27906 n r 2 - 109806 n s - 27906 r s + 40950 s + 151811 n + 31031 r - 120780 s / 5 2 + 139530) A[n + 5](r, s) / ((n - s + 5) (n + r + 6)) - (s + 1) / 12 11 10 2 9 3 8 4 (n - s + 6) (15 n - 120 n s + 420 n s - 840 n s + 1050 n s 7 5 6 6 5 7 4 8 11 10 - 840 n s + 420 n s - 120 n s + 15 n s + 960 n - 7080 n s 9 2 8 3 7 4 6 5 5 6 + 22680 n s - 41160 n s + 46200 n s - 32760 n s + 14280 n s 4 7 3 8 10 9 8 2 - 3480 n s + 360 n s + 28080 n - 189320 n s + 549495 n s 7 3 6 4 5 5 4 6 3 7 - 893730 n s + 886825 n s - 546180 n s + 201945 n s - 40370 n s 2 8 9 8 7 2 6 3 + 3255 n s + 496360 n - 3028530 n s + 7865610 n s - 11285840 n s 5 4 4 5 3 6 2 7 + 9698400 n s - 5045010 n s + 1519930 n s - 234060 n s 8 8 7 6 2 + 13140 n s + 5905455 n - 32202270 n s + 73661865 n s 5 3 4 4 3 5 2 6 - 91331796 n s + 66082990 n s - 27877330 n s + 6418965 n s 7 8 7 6 5 2 - 677860 n s + 19981 s + 49819350 n - 238969130 n s + 471576792 n s 4 3 3 4 2 5 6 - 491174120 n s + 287246610 n s - 92132430 n s + 14416880 n s 7 6 5 4 2 - 783952 s + 305576380 n - 1262892888 n s + 2089984650 n s 3 3 2 4 5 6 - 1755291800 n s + 777767800 n s - 168592640 n s + 13448498 s 5 4 3 2 2 3 + 1373092584 n - 4752891980 n s + 6331563440 n s - 4019285000 n s 4 5 4 3 + 1199270880 n s - 131749924 s + 4486031425 n - 12483661220 n s 2 2 3 4 3 + 12548040870 n s - 5350818724 n s + 806195025 s + 10392588740 n 2 2 3 2 - 21793731260 n s + 14690000668 n s - 3155387364 s + 16205347360 n 2 - 22760092352 n s + 7714449088 s + 15271736856 n - 10772151512 s / 4 5 + 6577964656) A[n + 5](r - 1, s + 1) / (x (n - s + 4) (n - s + 5) / 2 (n + r + 6)) + A[n + 6](r, s) + (s + 1) (2 n - 2 n s + 24 n - 13 s + 71) ( 4 3 2 2 3 2 2 2 3 n - 6 n s + 3 n s + 72 n - 111 n s + 39 n s + 645 n - 681 n s 2 4 3 2 2 3 2 + 127 s + 2556 n - 1384 s + 3781) (n - 2 n s + n s + 24 n - 37 n s 2 2 2 + 13 n s + 215 n - 227 n s + 43 s + 852 n - 460 s + 1261) / 5 A[n + 6](r - 1, s + 1) / (x (n - s + 5) (n + r + 6)) / 6 (n + 7) A[n + 7](r - 1, s + 1) - ------------------------------- = 0 x (n + r + 6) replacing n by, n - 7, changing variables, and moving a[n](r,s) to the left \ side, yields the statement of the thereom. QED. ------------------------------------------------- This took, 35.985, seconds. -------------------------