Theorem: For any integers p and q define the Abel-sum type sequence by n ----- \ k (k - 1 + p) (n - k + q) A[n](r, s) = ) binomial(n, k) x (r + k) (s - 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](r, s) = (n x + r x) A[n - 1](r + 1, s - 1) + s A[n - 1](r, s) + (-n r x - n s x + r x + s x) A[n - 2](r + 1, s - 1) and in Maple notation A[n](r,s) = (n*x+r*x)*A[n-1](r+1,s-1)+s*A[n-1](r,s)+(-n*r*x-n*s*x+r*x+s*x)*A[n-\ 2](r+1,s-1) ------------------------------------------------- This took, 0.123, seconds. ---------------------------------------------------------------- Theorem: For any integers p and q define the Abel-sum type sequence by n ----- \ 2 k (k - 1 + p) (n - k + q) A[n](r, s) = ) binomial(n, k) x (r + k) (s - k) / ----- k = 0 Then we have the following functional recurrence relating , A[n](r, s), A[n + 1](r, s), A[n + 2](r, s), A[n + 3](r, s) 2 3 (r + s) (n + 1) (n + 2 r - 1) (n + r - s - 3) x A[n](r, s) - ------------------------------------------------------------ - 2 (r + s) 2 (n - 2 s - 3) (s + 3) (n + r - s - 1) 2 2 2 2 2 (n + 1) (n + 2 n r - 2 n s + r - 2 r s + s - 4 n - 5 r + 3 s + 3) x / 2 A[n](r - 1, 1 + s) / ((n - 2 s - 3) (s + 3) (n + r - s - 1)) / 2 x (r + s) (n + 1) A[n](r - 2, s + 2) - ------------------------------------- 2 (s + 3) 3 2 x (n + 2 r - 1) (n + r + 1) (n + r - s - 3) A[n + 1](r, s) 4 + ------------------------------------------------------------ + (2 n 2 (n - 2 s - 3) (s + 3) (n + r - s - 1) 3 3 2 2 2 3 2 2 + 10 n r - 2 n s + 17 n r - 10 n r s + 9 n r - 17 n r s + 2 n r s 3 2 2 3 2 2 2 2 - 6 r s + 6 r s - 8 n - 28 n r + 6 n s - 24 n r + 22 n r s - 6 n s 3 2 2 2 2 2 + 5 r + 23 r s - 14 r s + 7 n + 3 n r - 21 n s - 19 r - 46 r s + 2 s 2 / - 4 n - 7 r + 7 s + 3) x A[n + 1](r - 1, 1 + s) / ((n - 2 s - 3) / 2 4 3 3 2 2 2 (s + 3) (n + r - s - 1)) + (n + 5 n r - n s + 4 n r - 14 n r s 2 2 2 2 3 2 2 3 3 - 3 n s - 10 n r s + 13 n r s + 3 n s + 6 r s - 6 r s - 6 n 2 2 2 2 2 2 - 21 n r + 13 n s - 12 n r + 40 n r s - 24 n s + 10 r s - 37 r s 3 2 2 2 + 17 s + 20 n + 25 n r - 73 n s - 2 r - 66 r s + 61 s - 42 n - 33 r / 2 + 69 s + 27) x A[n + 1](r - 2, s + 2) / ((n - 2 s - 3) (s + 3) / (n + r - s - 1)) + A[n + 1](r - 3, s + 3) - 2 (n + r + 1) 2 2 / (n + 2 n r + 2 n + 5 r - 1) (n + r - s - 3) x A[n + 2](r - 1, 1 + s) / / 2 4 3 ((n + 2) (n - 2 s - 3) (s + 3) (n + r - s - 1)) - 2 x (n + 2 n r 3 2 2 2 2 2 3 2 2 2 - 2 n s + n r - 2 n r s + n s - n + n r - 3 n s + 2 n r 2 2 2 2 - 8 n r s + 2 n s - 8 n - 15 n r + 3 n s - r - 8 r s - s - 15 r - 3 s) / 2 A[n + 2](r - 2, s + 2) / ((n + 2) (n - 2 s - 3) (s + 3) (n + r - s - 1)) / 2 2 (n - 2 n s - n - 5 s - 9) A[n + 2](r - 3, s + 3) - --------------------------------------------------- (n + 2) (s + 3) (n - 2 s - 3) x (n + 3) (n + 2 r) (n + r - s - 3) A[n + 3](r - 2, s + 2) + ---------------------------------------------------------- 2 (n + 2) (n - 2 s - 3) (s + 3) (n + r - s - 1) (n + 3) (n - 2 s - 4) A[n + 3](r - 3, s + 3) + -------------------------------------------- = 0 2 (n - 2 s - 3) (n + 2) (s + 3) and in Maple notation -(r+s)*(n+1)^2*(n+2*r-1)*(n+r-s-3)*x^3/(n-2*s-3)/(s+3)^2/(n+r-s-1)*A[n](r,s)-2* (r+s)*(n+1)^2*(n^2+2*n*r-2*n*s+r^2-2*r*s+s^2-4*n-5*r+3*s+3)*x^2/(n-2*s-3)/(s+3) ^2/(n+r-s-1)*A[n](r-1,1+s)-x*(r+s)*(n+1)^2/(s+3)^2*A[n](r-2,s+2)+x^3*(n+2*r-1)* (n+r+1)^2*(n+r-s-3)/(n-2*s-3)/(s+3)^2/(n+r-s-1)*A[n+1](r,s)+(2*n^4+10*n^3*r-2*n ^3*s+17*n^2*r^2-10*n^2*r*s+9*n*r^3-17*n*r^2*s+2*n*r*s^2-6*r^3*s+6*r^2*s^2-8*n^3 -28*n^2*r+6*n^2*s-24*n*r^2+22*n*r*s-6*n*s^2+5*r^3+23*r^2*s-14*r*s^2+7*n^2+3*n*r -21*n*s-19*r^2-46*r*s+2*s^2-4*n-7*r+7*s+3)*x^2/(n-2*s-3)/(s+3)^2/(n+r-s-1)*A[n+ 1](r-1,1+s)+(n^4+5*n^3*r-n^3*s+4*n^2*r^2-14*n^2*r*s-3*n^2*s^2-10*n*r^2*s+13*n*r *s^2+3*n*s^3+6*r^2*s^2-6*r*s^3-6*n^3-21*n^2*r+13*n^2*s-12*n*r^2+40*n*r*s-24*n*s ^2+10*r^2*s-37*r*s^2+17*s^3+20*n^2+25*n*r-73*n*s-2*r^2-66*r*s+61*s^2-42*n-33*r+ 69*s+27)*x/(n-2*s-3)/(s+3)^2/(n+r-s-1)*A[n+1](r-2,s+2)+A[n+1](r-3,s+3)-2*(n+r+1 )*(n^2+2*n*r+2*n+5*r-1)*(n+r-s-3)*x^2/(n+2)/(n-2*s-3)/(s+3)^2/(n+r-s-1)*A[n+2]( r-1,1+s)-2*x*(n^4+2*n^3*r-2*n^3*s+n^2*r^2-2*n^2*r*s+n^2*s^2-n^3+n^2*r-3*n^2*s+2 *n*r^2-8*n*r*s+2*n*s^2-8*n^2-15*n*r+3*n*s-r^2-8*r*s-s^2-15*r-3*s)/(n+2)/(n-2*s-\ 3)/(s+3)^2/(n+r-s-1)*A[n+2](r-2,s+2)-2*(n^2-2*n*s-n-5*s-9)/(n+2)/(s+3)/(n-2*s-3 )*A[n+2](r-3,s+3)+x*(n+3)*(n+2*r)*(n+r-s-3)/(n+2)/(n-2*s-3)/(s+3)^2/(n+r-s-1)*A [n+3](r-2,s+2)+(n+3)*(n-2*s-4)/(n-2*s-3)/(n+2)/(s+3)^2*A[n+3](r-3,s+3) = 0 ---------------------------------------------------------------- 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, 0.110, 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, 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) ------------------------------------------------- This took, 0.552, seconds. ---------------------------------------------------------------- ------------------------ All these took, 45.396, seconds