Theorem: Let F(m,n) be the coefficient of x^m*y^n in the Maclaurin series of 1 ------------------------------- 2 2 1 - x - y + 5 x y + 5 x + 2 y F(m,n) satisfies the following pure recurrences (39 + 9 n + 10 m) (n + 3 + m) (n + 2 + m) F(m, n) 75/7 ------------------------------------------------- (m + 3) (m + 4) (10 m + 29 + 9 n) 2 2 (n + 3 + m) (18 n + 42 n + 11 m n - 66 - 10 m - 49 m) F(m + 1, n) + 5/7 ------------------------------------------------------------------- (m + 4) (m + 3) (10 m + 29 + 9 n) 2 3 2 + 1/7 (15186 + 10965 n + 2577 n + 198 n + 14578 m + 7030 m n + 832 m n 2 2 3 + 4612 m + 1112 m n + 480 m ) F(m + 2, n)/((m + 4) (m + 3) (10 m + 29 + 9 n)) 2 2 (644 + 427 n + 63 n + 355 m + 115 m n + 50 m ) F(m + 3, n) - 1/7 ----------------------------------------------------------- (m + 4) (10 m + 29 + 9 n) + F(m + 4, n) = 0 30 (n + 3 + m) (n + 2 + m) (45 n + 177 + 44 m) F(m, n) -- --------------------------------------------------- - 5/19 (n + 3 + m) 19 (m + 4) (m + 3) (44 m + 133 + 45 n) 2 2 (180 n + 1482 n + 491 m n + 2886 + 308 m + 1943 m) F(m, n + 1)/((m + 4) 2 3 (m + 3) (44 m + 133 + 45 n)) + 1/19 (90258 + 65904 n + 15315 n + 1125 n 2 2 2 3 + 85847 m + 42065 m n + 4970 m n + 26903 m + 6619 m n + 2772 m ) F(m, n + 2)/((m + 4) (m + 3) (44 m + 133 + 45 n)) 2 2 (14668 + 7427 n + 855 n + 8741 m + 2141 m n + 1276 m ) F(m, n + 3) - 1/19 ------------------------------------------------------------------- (m + 4) (44 m + 133 + 45 n) + F(m, n + 4) = 0 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\ := 75 (39 + 9 n + 10 m) (n + 3 + m) (n + 2 + m) -------------------------------------------- 7 (m + 3) (m + 4) (10 m + 29 + 9 n) 2 2 5 (n + 3 + m) (18 n + 42 n + 11 m n - 66 - 10 m - 49 m) M + ----------------------------------------------------------- + (15186 7 (m + 4) (m + 3) (10 m + 29 + 9 n) 2 3 2 2 + 10965 n + 2577 n + 198 n + 14578 m + 7030 m n + 832 m n + 4612 m 2 3 2 + 1112 m n + 480 m ) M /(7 (m + 4) (m + 3) (10 m + 29 + 9 n)) 2 2 3 (644 + 427 n + 63 n + 355 m + 115 m n + 50 m ) M 4 - -------------------------------------------------- + M 7 (m + 4) (10 m + 29 + 9 n) or equivalenty 4 2 3 17550 + 2436 M + 6675 m + 750 m - 990 M + 18675 n + 19125 m + 130 M m n 2 2 2 2 2 2 2 + 145 M m n + 5 M m n + 7030 M m n + 832 M m n + 1112 M m n 3 3 2 3 2 4 4 2 - 772 M m n - 63 M m n - 115 M m n + 441 M m n + 63 M m n 2 2 3 2 2 3 - 395 m M + 6300 n - 1932 M + 15186 M + 14578 m M - 50 m M 3 2 2 4 4 3 3 3 3 4 + 480 m M + 693 m M + 2261 m M - 50 m M - 1709 m M + 70 m M 2 3 2 2 2 2 2 2 3 3 - 505 m M + 4612 m M + 10965 M n + 2577 M n + 198 M n - 1281 M n 3 2 4 2 3 3 - 189 M n + 756 M n + 300 M n + 480 M n + 90 M n + 675 n + 12975 m n 2 2 + 2100 m n + 2175 m n - 1065 m M 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 + 5 M N + 5 N + 2 M The sequence of successive commutators is 2 2 2 [(20250 n + 166500 n + 42000 m n + 339750 + 171750 m + 21750 m ) N 2 2 + (347250 + 22125 m + 169875 n + 175125 m + 20625 n + 42750 m n) M N 2 2 2 + (-6550 m - 9850 m n - 41850 n - 3450 n - 54450 m - 117000) M N 2 2 2 + (8400 n + 69300 n + 17400 m n + 141900 + 71400 m + 9000 m ) M 2 2 2 + (-4150 n - 11400 m n - 49400 n - 62750 m - 7400 m - 137250) M N 2 2 2 2 + (33150 m n + 20110 m + 144455 n + 13505 n + 391530 + 174470 m) M N 2 2 3 + (-8220 - 600 m + 560 n - 4360 m + 580 n + 40 m n) M 2 2 3 + (214590 + 19130 m n + 75680 n + 6570 n + 13055 m + 105635 m) M N 2 2 3 2 + (-6212 m n - 5104 m - 23494 n - 1190 n - 88844 - 42780 m) M N 2 2 4 + (102888 + 5760 m + 37016 n + 48416 m + 3328 n + 8896 m n) M 2 2 4 + (-108776 - 7892 m n - 34539 n - 2573 n - 5317 m - 47643 m) M N 2 2 4 2 + (8258 m n + 6114 m + 39743 n + 2915 n + 144106 + 58318 m) M N 2 2 5 + (-600 m - 5240 m - 920 m n - 4008 n - 11676 - 252 n ) M 2 2 5 + (30858 + 1216 m n + 5154 n + 126 n + 1780 m + 14730 m) M N 2 2 5 2 + (-866 m - 838 m n - 8066 m - 3768 n - 19212 - 126 n ) M N 2 6 + (15708 + 504 m n + 840 m + 7224 m + 2268 n) M 2 6 + (-9744 - 252 m n - 483 m - 4305 m - 1134 n) M N 2 6 2 + (11634 + 252 m n + 546 m + 4998 m + 1134 n) M N , 4 3 8 (-2095080 - 391380 m - 272540 n) M N + (-42168 - 2016 n - 7728 m) M N 6 2 + (94334 n + 848848 + 150864 m) M N 3 + (825000 n + 855000 m + 4222500) M N 4 + (-620400 - 91200 n - 118400 m) M N 4 4 8 4 + (189592 n + 251952 m + 1472696) M N + (15456 + 2688 m + 504 n) M N 8 2 4 + (2520 n + 60900 + 10920 m) M N + (-192000 n - 226000 m - 1179000) M N 4 2 + (457650 n + 3049320 + 588960 m) M N 4 + (405000 n + 420000 m + 2070000) N 5 + (1053960 + 153040 n + 208880 m) M N 7 2 + (-10648 n - 153546 - 28282 m) M N 5 2 + (-166286 n - 1297724 - 245644 m) M N 2 2 5 + (3874500 + 783000 m + 756000 n) M N + (-22240 + 160 n - 4800 m) M 2 3 + (-487500 m - 416500 n - 2564000) M N 5 3 + (1439648 + 258116 m + 170488 n) M N 2 4 + (535300 m + 458750 n + 2905350) M N 7 3 8 + (119632 + 22044 m + 6376 n) M N + (35616 + 2016 n + 6720 m) M 6 4 8 3 + (29518 n + 46900 m + 295258) M N + (-27636 - 4872 m - 1008 n) M N 3 6 + (354000 m + 1755000 + 342000 n) M N + (239744 + 35584 n + 46080 m) M 5 4 6 + (-39622 n - 69964 m - 394866) M N + (-466216 - 63136 n - 85072 m) M N 7 4 6 3 + (-2432 n - 6736 m - 39304) M N + (-537172 - 91128 m - 55592 n) M N 7 4 + (-25760 - 3680 n - 4800 m) M + (357600 + 69600 n + 72000 m) M 3 2 + (-1930650 - 367050 m - 304050 n) M N 7 + (146320 + 9728 n + 28480 m) M N 3 4 + (-214540 m - 144470 n - 1134550) M N 3 3 + (805800 m + 654800 n + 4257100) M N , 6 2 2 2 2 2 2 6 2 (38 M N - 28 M N + 90 N + 95 M N - 29 M N + 40 M ) (28 M N 5 2 2 4 2 3 2 2 2 2 - 48 M N + 433 N M - 222 N M + 1305 M N - 400 M N + 1500 N 6 5 4 3 2 6 - 28 M N + 34 M N - 398 M N + 690 M N - 550 M N + 1500 M N + 56 M 5 4 3 2 - 40 M + 384 M - 40 M + 600 M ) 2 2 2 2 2 2 (M N - M N - M N + 5 M N + 5 N + 2 M )] As you can see, the last entry, 6 2 2 2 2 2 2 6 2 (38 M N - 28 M N + 90 N + 95 M N - 29 M N + 40 M ) (28 M N 5 2 2 4 2 3 2 2 2 2 - 48 M N + 433 N M - 222 N M + 1305 M N - 400 M N + 1500 N 6 5 4 3 2 6 - 28 M N + 34 M N - 398 M N + 690 M N - 550 M N + 1500 M N + 56 M 5 4 3 2 - 40 M + 384 M - 40 M + 600 M ) 2 2 2 2 2 2 (M N - M N - M N + 5 M N + 5 N + 2 M ), is a multiple of , 2 2 2 2 2 2 M N - M N - M N + 5 M N + 5 N + 2 M The proof follows by backwards induction, after checking the boundary condit\ ions QED . ----------------------------------------------------------------------------\ ---- The whole thing took , 19.830, seconds of CPU time