Theorem: Let F(m,n) be the coefficient of x^m*y^n in the Maclaurin series of 1 ----------------------- 2 2 1 - x - y + 2 x + 3 y F(m,n) satisfies the following pure recurrences 48 (n + m + 3) (n + m + 2) F(m, n) -- ------------------------------- 11 (m + 4) (m + 2) 2 12 (2 m n + 5 n + 26 + 4 m + 21 m) (n + m + 3) F(m + 1, n) - -- -------------------------------------------------------- 11 (m + 4) (m + 3) (m + 2) 2 2 (606 + 171 n + 3 n + 376 m + 56 m n + 58 m ) F(m + 2, n) + 1/11 --------------------------------------------------------- (m + 3) (m + 4) (86 + 11 n + 23 m) F(m + 3, n) - 1/11 ------------------------------ + F(m + 4, n) = 0 m + 4 (n + m + 3) (n + m + 2) F(m, n) 72/7 ------------------------------- (m + 4) (m + 2) 2 (2 m n + 5 n + 26 + 4 m + 21 m) (n + m + 3) F(m, n + 1) - 12/7 -------------------------------------------------------- (m + 4) (m + 3) (m + 2) 2 2 (552 + 152 n + 2 n + 343 m + 50 m n + 53 m ) F(m, n + 2) + 1/7 --------------------------------------------------------- (m + 3) (m + 4) (56 + 7 n + 15 m) F(m, n + 3) - 1/7 ----------------------------- + F(m, n + 4) = 0 m + 4 Proof: We will only do the first recurrence Let M and N be the shift operators in the m and n directions, respectively (M f(m,n):=f(m+1,n), Nf(m,n):=f(m,n+1) ) We have to prove that F(m,n) is annihilated by the operator, let's call it Q\ := 2 48 (n + m + 3) (n + m + 2) 12 (2 m n + 5 n + 26 + 4 m + 21 m) (n + m + 3) M -------------------------- - ------------------------------------------------- 11 (m + 4) (m + 2) 11 (m + 4) (m + 3) (m + 2) 2 2 2 3 (606 + 171 n + 3 n + 376 m + 56 m n + 58 m ) M (86 + 11 n + 23 m) M + ------------------------------------------------ - --------------------- 11 (m + 3) (m + 4) 11 (m + 4) 4 + M or equivalenty 2 2 2 2 2 2 3 864 + 96 m n + 48 n m + 342 M n + 492 m M + 1358 m M - 48 m M 2 2 2 2 2 2 - 72 M m n - 24 M n m - 396 m M - 1068 m M + 6 M n - 60 M n 3 3 2 2 2 2 2 2 - 492 M n - 55 M m n - 11 M m n + 56 M m n + 3 M n m + 283 M m n 3 4 2 4 4 3 3 2 3 3 + 11 m M + 99 m M + 286 m M - 23 m M - 201 m M - 568 m M 3 3 2 4 - 66 M n + 58 m M - 384 M m n + 264 M - 936 M + 720 n + 1008 m 3 2 3 2 2 - 516 M + 1212 M + 48 m + 384 m + 528 m n + 144 n We know, by the obvious algebra, that F(m,n) is annihilated by 2 2 2 2 2 2 M N - M N - M N + 2 N + 3 M The sequence of successive commutators is 2 [192 (m + 3) (2 m + 7 + 2 n) N 2 2 2 + (-48 n - 576 m n - 1872 n - 624 m - 4272 m - 7344) M N 2 2 2 + (1152 m n + 864 m + 6336 m + 288 n + 4320 n + 11808) M 2 2 2 + (-3072 m - 6336 - 480 m n - 1920 n - 96 n - 384 m ) M N 2 2 2 2 + (14376 + 992 m + 7372 m + 3240 n + 120 n + 840 m n) M N 2 2 3 + (-864 m n - 144 n - 3168 n - 6480 m - 12312 - 864 m ) M 2 2 3 + (336 m n + 48 n + 1272 n + 2856 m + 5760 + 360 m ) M N 2 2 3 2 + (-11202 - 762 m - 5780 m - 1863 n - 51 n - 508 m n) M N 2 2 4 + (7992 m + 672 m n + 1044 m + 15444 + 2370 n + 18 n ) M 2 2 4 + (-3174 m - 230 m n - 404 m - 6292 - 814 n - 6 n ) M N 2 2 4 2 + (551 m + 258 m n + 4315 m + 904 n + 8516 + 6 n ) M N 2 5 + (-414 m - 3240 m - 132 m n - 6372 - 462 n) M 2 5 + (149 m + 1179 m + 44 m n + 2344 + 154 n) M N 2 5 2 2 6 + (-193 m - 44 m n - 1509 m - 154 n - 2960) M N + 198 (m + 4) M 2 6 2 6 2 3 4 - 66 (m + 4) M N + 66 (m + 4) M N , (-3072 n - 9344 m - 42144) M N 4 6 + (48384 + 6912 n + 10368 m) M + (60480 + 4032 n + 12528 m) M 5 2 2 4 + (-96792 - 20064 m - 6864 n) M N + (8064 m + 34272 + 2496 n) M N 7 4 3 + (-24408 - 792 n - 4968 m) M + (-47504 - 9616 m - 3840 n) M N 4 8 3 + (-46080 - 9216 m - 5760 n) M N + (-528 m - 2640) M N 8 2 6 2 + (1848 m + 9240) M N + (74248 + 15070 m + 3568 n) M N 2 3 7 4 + (-3840 m - 1536 n - 16896) M N + (-4868 - 88 n - 992 m) M N 5 3 6 + (35076 + 7112 m + 2236 n) M N + (-47784 - 9696 m - 2760 n) M N 5 2 2 + (-49248 - 5184 n - 10368 m) M + (4608 n + 39168 + 9216 m) M N 7 2 3 3 + (-25936 - 5272 m - 616 n) M N + (2496 n + 6144 m + 28224) M N 7 3 4 4 + (8064 + 1632 m + 176 n) M N + (44184 + 2920 n + 9220 m) M N 6 4 4 2 + (15796 + 584 n + 3194 m) M N + (122928 + 25824 m + 11232 n) M N 4 5 + (1536 m + 4608) N + (42912 + 8640 m + 4032 n) M N 4 8 4 + (-768 n - 3840 m - 13824) M N + (264 m + 1320) M N 5 4 3 2 + (-28400 - 1502 n - 5858 m) M N + (-66240 - 14976 m - 6912 n) M N 8 6 3 + (2376 m + 11880) M + (-24520 - 4924 m - 1056 n) M N 7 8 + (17724 + 3576 m + 528 n) M N + (-1584 m - 7920) M N, 6 M 2 2 6 2 5 2 2 4 2 3 (2 M N - N - 2 N M + 6 M) (44 M N - 136 M N + 413 N M - 616 N M 2 2 2 2 6 5 4 3 + 864 M N - 576 M N + 384 N - 44 M N + 114 M N - 344 M N + 336 M N 2 6 5 4 3 2 - 384 M N + 132 M - 276 M + 696 M - 576 M + 576 M ) 2 2 2 2 2 2 (M N - M N - M N + 2 N + 3 M )] 2 2 6 2 As you can see, the last entry, 6 M (2 M N - N - 2 N M + 6 M) (44 M N 5 2 2 4 2 3 2 2 2 2 - 136 M N + 413 N M - 616 N M + 864 M N - 576 M N + 384 N 6 5 4 3 2 6 5 - 44 M N + 114 M N - 344 M N + 336 M N - 384 M N + 132 M - 276 M 4 3 2 2 2 2 2 2 2 + 696 M - 576 M + 576 M ) (M N - M N - M N + 2 N + 3 M ), 2 2 2 2 2 2 is a multiple of , M N - M N - M N + 2 N + 3 M The proof follows by backwards induction, after checking the boundary condit\ ions QED . ----------------------------------------------------------------------------\ ---- The whole thing took , 19.061, seconds of CPU time