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 expressiing A[n](r.s) in te\ rms 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) ------------------------------------------------- This took, 0.124, 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 expressiing A[n](r.s) in te\ rms 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) ------------------------------------------------- This took, 0.116, 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 expressiing A[n](r.s) in te\ rms 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, -1 + s) 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, -1 + s) / ((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, -1 + s) / ((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, -1 + s) - ---------------------------------------------------------------------- 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,-1+s)+(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,-1+s) -(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,-1+s)+(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,-1+s) ------------------------------------------------- This took, 0.531, 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 expressiing A[n](r.s) in te\ rms 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 2 3 6 - 12 s ) A[n - 2](r, s) / (n (n - 2 - s) (n - 1 - 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 2 3 3 6 A[n - 3](r + 1, s - 1) / (n (n - 2 - s) (n - 1 - 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 - 3 - s) (n - 2 - s) (n - 1 - 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 - 6 r s ) A[n - 4](r + 1, s - 1) / (n (n - 3 - s) (n - 2 - s) / 4 3 2 2 3 2 s (n - 3 n s + 3 n s - s ) A[n - 4](r, s) (n - 1 - s) ) - --------------------------------------------- + x 4 n (n - 3 - s) (n - 2 - s) (n - 1 - 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 - 3 - s) (n - 2 - s) (n - 1 - 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-2-s)^2/(n-1-s)^3*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-2-s)^2/(n-1-s)^3*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-3-s)/(n-2-s)^2/(n-1-s)^2*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-3-s)/(n-2-s)^2/(n-1-s)^2*A[n-4](r+1,s-1)- s^4*(n^3-3*n^2*s+3*n*s^2-s^3)/n^4/(n-3-s)/(n-2-s)/(n-1-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-3-s)/(n-2-s)/(n-1 -s)*A[n-5](r+1,s-1) ------------------------------------------------- This took, 2.197, 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 expressiing A[n](r.s) in te\ rms 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, s - 1) 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, s - 1) / (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, s - 1) / (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 4 A[n - 4](r + 1, s - 1) / (n (n - 1 - s) (n - 2 - s) (n - 3 - s) ) - s / 8 7 6 2 5 3 4 4 3 5 2 6 (5 n - 35 n s + 105 n s - 175 n s + 175 n s - 105 n s + 35 n s 7 7 6 5 2 4 3 3 4 - 5 n s - 50 n + 310 n s - 810 n s + 1150 n s - 950 n s 2 5 6 7 6 5 4 2 + 450 n s - 110 n s + 10 s + 175 n - 945 n s + 2100 n s 3 3 2 4 5 6 5 4 - 2450 n s + 1575 n s - 525 n s + 70 s - 250 n + 1150 n s 3 2 2 3 4 5 4 3 - 2100 n s + 1900 n s - 850 n s + 150 s + 120 n - 456 n s 2 2 3 4 / 5 + 648 n s - 408 n s + 96 s ) A[n - 4](r, s) / (n (n - 4 - s) / 2 2 2 8 7 7 (n - 1 - s) (n - 2 - s) (n - 3 - s) ) + x (n + 5 n r - 3 n s 6 6 2 5 2 5 3 4 3 4 4 - 35 n r s - 7 n s + 105 n r s + 49 n s - 175 n r s - 105 n s 3 4 3 5 2 5 2 6 6 7 + 175 n r s + 119 n s - 105 n r s - 77 n s + 35 n r s + 27 n s 7 8 7 6 6 5 5 2 - 5 r s - 4 s - 10 n - 40 n r + 30 n s + 240 n r s + 30 n s 4 2 4 3 3 3 3 4 2 4 - 600 n r s - 250 n s + 800 n r s + 450 n s - 600 n r s 2 5 5 6 6 7 6 5 - 390 n s + 240 n r s + 170 n s - 40 r s - 30 s + 35 n + 105 n r 5 4 3 2 3 3 2 3 - 105 n s - 525 n r s + 1050 n r s + 350 n s - 1050 n r s 2 4 4 5 5 6 5 4 - 525 n s + 525 n r s + 315 n s - 105 r s - 70 s - 50 n - 100 n r 4 3 3 2 2 2 2 3 3 + 150 n s + 400 n r s - 100 n s - 600 n r s - 100 n s + 400 n r s 4 4 5 4 3 3 2 + 150 n s - 100 r s - 50 s + 24 n + 24 n r - 72 n s - 72 n r s 2 2 2 3 3 / 5 + 72 n s + 72 n r s - 24 n s - 24 r s ) A[n - 5](r + 1, s - 1) / (n / 2 2 2 (n - 4 - s) (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) 4 4 + ------------------------------------------------------- - x (n r + n s 5 n (n - 4 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) 3 3 2 2 2 2 3 3 4 4 5 - 4 n r s - 4 n s + 6 n r s + 6 n s - 4 n r s - 4 n s + r s + s / 5 ) A[n - 6](r + 1, s - 1) / (n (n - 4 - s) (n - 1 - s) (n - 2 - s) / (n - 3 - s)) and in Maple notation A[n](r,s) = x*(n+r)/n^5*A[n-1](r+1,s-1)+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,s-1)-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,s -1)+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,s-1)-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-4-s)/(n-1-s)^2/(n-2-s)^2/(n-3-s)^2*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-4-s)/(n-1-s)^2/(n-2-s)^2/(n-3-s)^2*A[n-5](r+1,s-1)+s^5*(n^4-4* n^3*s+6*n^2*s^2-4*n*s^3+s^4)/n^5/(n-4-s)/(n-1-s)/(n-2-s)/(n-3-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-4-s)/(n-1-s)/(n-2-s)/(n-3-s)*A[n-6](r+1,s-1) ------------------------------------------------- This took, 9.788, 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 expressiing A[n](r.s) in te\ rms 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 3 3 3 2 A[n - 4](r, s) / (n (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 4 - 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 3 3 3 ) A[n - 5](r + 1, s - 1) / (n (n - 1 - s) (n - 2 - s) (n - 3 - s) / 2 5 10 9 8 2 7 3 6 4 (n - 4 - 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 - 5 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 4 - 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 - 5 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 4 - 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 - 5 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 4 - 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 - 5 - s) (n - 1 - s) (n - 2 - s) (n - 3 - s) (n - 4 - 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 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*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-5-s)/(n-1-s)^2/(n-2-s)^2/(n-3-s)^2/(n-4-s)^2*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-5-s)/( n-1-s)/(n-2-s)/(n-3-s)/(n-4-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-5-s)/(n-1-s)/(n-2-s)/(n-3-s)/(n-4-s)*A[n-7](r+1,s-1) ------------------------------------------------- This took, 36.456, seconds. -------------------------