A Linear Recurrence Equation for The Diagonal Coefficients of The Power Seres Of 1 ---------------------------- 2 2 1/2 ((1 - x) + (1 - y) - 1) By Shalosh B. Ekhad n n Let , A(n), be the coefficient of, x y , in the Maclaurin Expansion of the Formal Power series, 1 ---------------------------- 2 2 1/2 ((1 - x) + (1 - y) - 1) then , A(n), satisfies the following linear recurrence equation with polynomi\ al coefficients 2 (2 n + 3) (2 n + 1) A(n) 3 (2 n + 3) A(n + 1) ------------------------ - --------------------- + A(n + 2) = 0 2 2 (n + 2) (n + 2) Proof: Thanks to Cauchy's Integral Formula, A(n) equals to a constant (independent of n) times the contour-integral with respect to the complex variables, x, y around any poly-circle around the origin, of the function 1 F(n, x, y) = ---------------------------------------------- 2 2 1/2 (n + 1) (n + 1) ((1 - x) + (1 - y) - 1) x y Let's cleverly constuct rational functions 2 2 2 3 2 3 R[1](n, x, y) = (-357 n x y + 60 x + 228 n x - 120 x - 90 x y - 30 n y 3 2 2 4 4 4 2 2 - 456 x n + 30 y x + 30 n y + 228 n x + 60 x + 129 n x y 2 4 2 2 3 3 2 3 3 2 + 51 n y - 622 x y n + 104 x y n - 4 y n + 64 n x y + 88 x y n 3 2 2 2 4 3 4 3 + 64 x y n - 34 y n + 48 x y n + 136 x y n + 40 n x y 2 3 2 4 2 2 2 2 2 3 2 2 4 - 491 x y n - 136 x y n + 157 y x n + 30 y x n - 16 y x n 3 4 3 2 3 2 4 2 3 3 3 + 40 x y n - 4 y n - 99 y n - 53 y n - 21 n y + 133 n y 4 3 3 4 4 4 2 2 2 3 2 4 + 88 n y - 42 n y - 39 n y + 303 x n + 168 x n + 33 x n 3 2 3 3 3 4 4 4 4 3 4 2 - 602 x n - 330 x n - 64 x n + 31 x n + 162 x n + 299 x n ) / 2 2 2 / (4 x y (n + 4 n + 4) (n + 1) n ) / 2 2 2 2 3 R[2](n, x, y) = - (-60 n x y + 90 x n y + 30 n x - 60 n y + 30 x 3 3 4 4 4 2 3 + 120 n y + 39 x n - 60 n y - 69 n x - 30 x + 8 n + 12 n 2 4 2 2 3 3 4 3 3 3 2 - 192 n y - 258 x y n + 64 x y n + 4 n + 104 n x y + 64 y x n 2 2 4 3 2 3 2 4 3 4 - 220 y n + 40 n x y - 358 x y n - 136 x y n + 40 x y n 3 2 2 2 2 3 2 4 2 3 3 3 + 77 x n y + 179 x n y - 201 y n - 53 y n + 404 n y + 348 n y 4 3 3 4 4 4 2 2 2 3 2 4 + 88 n y - 159 n y - 39 n y + 99 x n + 102 x n + 33 x n 3 2 3 3 3 4 4 4 4 3 4 2 / - 98 x n - 171 x n - 64 x n + 31 x n + 69 x n - x n ) / (4 y / 2 2 2 x (n + 4 n + 4) (n + 1) n ) With the motive that 2 (2 n + 3) (2 n + 1) F(n, x, y) 3 (2 n + 3) F(n + 1, x, y) ------------------------------ - --------------------------- + F(n + 2, x, y) 2 2 (n + 2) (n + 2) R[1](n, x, y) = D[x](----------------------------------------------) 2 2 1/2 (n + 1) (n + 1) ((1 - x) + (1 - y) - 1) x y R[2](n, x, y) + D[y](----------------------------------------------) 2 2 1/2 (n + 1) (n + 1) ((1 - x) + (1 - y) - 1) x y (Check!) and the theorem follows upon contour-integrating with respect to, x, y QED! This took, 0.604, seconds .