A Linear Recurrence Equation for The Diagonal Coefficients of The Power Ser\ ies Of 1 -------------------------------------------------------- (-2 x y z - x y - x z - y z + 1) (1 - x) (1 - y) (1 - z) By Shalosh B. Ekhad n n n Let , A(n), be the coefficient of, x y z , in the Maclaurin Expansion of the Formal Power series, 1 -------------------------------------------------------- (-2 x y z - x y - x z - y z + 1) (1 - x) (1 - y) (1 - z) then , A(n), satisfies the following linear recurrence equation with polynomi\ al coefficients 2 4 3 2 72 (n + 3) (n + 2) (245 n + 4074 n + 25199 n + 68630 n + 69360) A(n) ------------------------------------------------------------------------ - (6 + n) (n + 5) (n + 4) %1 6 5 4 3 2 (n + 3) (21805 n + 461811 n + 3994713 n + 18027793 n + 44669582 n 7 + 57481472 n + 29924736) A(n + 1)/((6 + n) (n + 5) (n + 4) %1) - (13230 n 6 5 4 3 2 + 342251 n + 3774381 n + 22986523 n + 83432029 n + 180334002 n + 214738496 n + 108574656) A(n + 2)/((6 + n) (n + 5) (n + 4) %1) + ( 7 6 5 4 3 21560 n + 564557 n + 6278499 n + 38414073 n + 139540493 n 2 + 300684010 n + 355520264 n + 177724704) A(n + 3)/((6 + n) (n + 5) 6 5 4 3 2 (n + 4) %1) - (4410 n + 96607 n + 869788 n + 4117797 n + 10803710 n + 14878592 n + 8390976) A(n + 4)/((6 + n) (n + 4) %1) + A(n + 5) = 0 4 3 2 %1 := 245 n + 3094 n + 14447 n + 29474 n + 22100 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, z around any poly-circle around the origin, of the function F(n, x, y, z) = 1/((-2 x y z - x y - x z - y z + 1) (1 - x) (1 - y) (1 - z) (n + 1) (n + 1) (n + 1) x y z ) Let's cleverly constuct rational function 38 11 4 R(y[1], y[2], z) = -1/497664 (57184244728380 n z y[1] 38 11 3 38 11 2 2 + 15652457929080 n z y[1] y[2] - 288560165286960 n z y[1] y[2] 38 11 3 38 11 4 + 15652457929080 n z y[1] y[2] + 57184244728380 n z y[2] 38 10 5 38 10 4 - 10990092839040 n z y[1] + 339383874627000 n z y[1] y[2] 38 10 3 2 - 176978321173080 n z y[1] y[2] 38 10 2 3 38 10 4 - 176978321173080 n z y[1] y[2] + 339383874627000 n z y[1] y[2] 38 10 5 38 9 6 - 10990092839040 n z y[2] + 10082015697600 n z y[1] 38 9 5 38 9 5 + 61273629375840 n z y[1] y[2] + 61273629375840 n z y[1] y[2] 38 9 6 38 8 7 + 10082015697600 n z y[2] - 20124253865520 n z y[1] 38 8 6 38 8 6 + 12924067257360 n z y[1] y[2] + 12924067257360 n z y[1] y[2] 38 8 7 38 7 8 - 20124253865520 n z y[2] + 20124253865520 n z y[1] 38 7 7 38 7 7 - 6708084621840 n z y[1] y[2] - 6708084621840 n z y[1] y[2] 38 7 8 38 6 9 + 20124253865520 n z y[2] - 10082015697600 n z y[1] 38 6 8 38 6 8 - 2855310736320 n z y[1] y[2] - 2855310736320 n z y[1] y[2] 38 6 9 38 5 10 - 10082015697600 n z y[2] + 10990092839040 n z y[1] 38 5 9 38 5 9 - 68297665554720 n z y[1] y[2] - 68297665554720 n z y[1] y[2] 38 5 10 38 4 11 + 10990092839040 n z y[2] - 57184244728380 n z y[1] 38 4 10 38 4 9 2 - 316659095437860 n z y[1] y[2] + 530934963519240 n z y[1] y[2] 38 4 2 9 38 4 10 + 530934963519240 n z y[1] y[2] - 316659095437860 n z y[1] y[2] 38 4 11 38 3 11 - 57184244728380 n z y[2] - 34713872838540 n z y[1] y[2] 38 3 10 2 - 124028316335700 n z y[1] y[2] 38 3 2 10 38 3 11 - 124028316335700 n z y[1] y[2] - 34713872838540 n z y[1] y[2] 38 2 11 2 + 144280082643480 n z y[1] y[2] 38 2 10 3 38 2 9 4 + 301006637508780 n z y[1] y[2] - 530934963519240 n z y[1] y[2] 38 2 4 9 38 2 3 10 - 530934963519240 n z y[1] y[2] + 301006637508780 n z y[1] y[2] 38 2 2 11 38 11 3 + 144280082643480 n z y[1] y[2] + 19061414909460 n z y[1] y[2] 38 10 4 38 9 5 - 22724779189140 n z y[1] y[2] + 7024036178880 n z y[1] y[2] 38 8 6 38 7 7 - 10068756521040 n z y[1] y[2] + 13416169243680 n z y[1] y[2] 38 6 8 38 5 9 - 10068756521040 n z y[1] y[2] + 7024036178880 n z y[1] y[2] 38 4 10 38 3 11 - 22724779189140 n z y[1] y[2] + 19061414909460 n z y[1] y[2] 38 11 3 38 11 2 - 53775287748000 n z y[1] + 93913751875860 n z y[1] y[2] 38 11 2 38 11 3 + 93913751875860 n z y[1] y[2] - 53775287748000 n z y[2] 38 10 4 38 10 2 2 + 521849843800740 n z y[1] + 2271749475046380 n z y[1] y[2] 38 10 4 38 9 5 + 521849843800740 n z y[2] + 46365222995280 n z y[1] 38 9 4 38 9 3 2 - 213282349915320 n z y[1] y[2] + 176978321173080 n z y[1] y[2] 38 9 2 3 38 9 4 + 176978321173080 n z y[1] y[2] - 213282349915320 n z y[1] y[2] 38 9 5 38 8 6 + 46365222995280 n z y[2] - 6947506026720 n z y[1] 38 8 5 38 8 5 + 33420334293360 n z y[1] y[2] + 33420334293360 n z y[1] y[2] 38 8 6 38 7 7 - 6947506026720 n z y[2] + 6708084621840 n z y[1] 38 7 6 38 7 6 - 6432666821280 n z y[1] y[2] - 6432666821280 n z y[1] y[2] 38 7 7 38 6 8 + 6708084621840 n z y[2] - 3121250494320 n z y[1] 38 6 7 38 6 7 + 6432666821280 n z y[1] y[2] + 6432666821280 n z y[1] y[2] 38 6 8 38 5 9 - 3121250494320 n z y[2] - 39341186816400 n z y[1] 38 5 8 38 5 8 - 29756970013680 n z y[1] y[2] - 29756970013680 n z y[1] y[2] 38 5 9 38 4 10 - 39341186816400 n z y[2] - 544574622989880 n z y[1] 38 4 9 38 4 9 + 58236471301680 n z y[1] y[2] + 58236471301680 n z y[1] y[2] 38 4 10 38 3 11 - 544574622989880 n z y[2] + 72836702657460 n z y[1] 38 3 9 2 38 3 2 9 - 176978321173080 n z y[1] y[2] - 176978321173080 n z y[1] y[2] 38 3 11 38 2 11 + 72836702657460 n z y[2] + 14644640774610 n z y[1] y[2] 38 2 10 2 - 1135874737523190 n z y[1] y[2] 38 2 2 10 38 2 11 - 1135874737523190 n z y[1] y[2] + 14644640774610 n z y[1] y[2] 38 11 2 38 9 4 - 108558392650470 n z y[1] y[2] + 155045878613640 n z y[1] y[2] 38 8 5 38 5 8 - 3663364279680 n z y[1] y[2] - 3663364279680 n z y[1] y[2] 38 4 9 38 2 11 + 155045878613640 n z y[1] y[2] - 108558392650470 n z y[1] y[2] 38 11 3 38 10 4 - 19061414909460 n y[1] y[2] + 22724779189140 n y[1] y[2] 38 9 5 38 8 6 - 7024036178880 n y[1] y[2] + 10068756521040 n y[1] y[2] 38 7 7 38 6 8 - 13416169243680 n y[1] y[2] + 10068756521040 n y[1] y[2] 38 5 9 38 4 10 - 7024036178880 n y[1] y[2] + 22724779189140 n y[1] y[2] 38 3 11 37 11 4 - 19061414909460 n y[1] y[2] + 20609248403715426 n z y[1] 37 11 3 - 1988542386498444 n z y[1] y[2] 37 11 2 2 - 44757976003288632 n z y[1] y[2] 37 11 3 37 11 4 - 1988542386498444 n z y[1] y[2] + 20609248403715426 n z y[2] 37 10 5 37 10 4 - 1651760992512288 n z y[1] + 67594364770031820 n z y[1] y[2] 37 10 3 2 - 34765779349573356 n z y[1] y[2] 37 10 2 3 - 34765779349573356 n z y[1] y[2] 37 10 4 37 10 5 + 67594364770031820 n z y[1] y[2] - 1651760992512288 n z y[2] 37 9 6 37 9 5 + 1418424999233760 n z y[1] + 9287058548077968 n z y[1] y[2] 37 9 4 2 37 9 2 4 - 1471314862080 n z y[1] y[2] - 1471314862080 n z y[1] y[2] 37 9 5 37 9 6 + 9287058548077968 n z y[1] y[2] + 1418424999233760 n z y[2] 37 8 7 37 8 6 - 2751888886634664 n z y[1] + 1644743121880152 n z y[1] y[2] 37 8 6 37 8 7 + 1644743121880152 n z y[1] y[2] - 2751888886634664 n z y[2] 37 7 8 37 7 7 + 2731764632769144 n z y[1] - 957544803275928 n z y[1] y[2] 37 7 7 37 7 8 - 957544803275928 n z y[1] y[2] + 2731764632769144 n z y[2] 37 6 9 37 6 8 - 1388178952140960 n z y[1] - 185154660759264 n z y[1] y[2] 37 6 8 37 6 9 - 185154660759264 n z y[1] y[2] - 1388178952140960 n z y[2] 37 5 10 37 5 9 + 1596810528317088 n z y[1] - 10083373490174544 n z y[1] y[2] 37 5 8 2 37 5 2 8 + 1471314862080 n z y[1] y[2] + 1471314862080 n z y[1] y[2] 37 5 9 37 5 10 - 10083373490174544 n z y[1] y[2] + 1596810528317088 n z y[2] 37 4 11 37 4 10 - 20208958690616766 n z y[1] - 58087867540691562 n z y[1] y[2] 37 4 9 2 + 104292433665846468 n z y[1] y[2] 37 4 2 9 + 104292433665846468 n z y[1] y[2] 37 4 10 37 4 11 - 58087867540691562 n z y[1] y[2] - 20208958690616766 n z y[2] 37 3 11 - 4660680343426758 n z y[1] y[2] 37 3 10 2 - 26850485463336990 n z y[1] y[2] 37 3 2 10 - 26850485463336990 n z y[1] y[2] 37 3 11 - 4660680343426758 n z y[1] y[2] 37 2 11 2 + 21080467257852996 n z y[1] y[2] 37 2 10 3 + 60076409927190006 n z y[1] y[2] 37 2 9 4 37 2 8 5 - 103229092423945908 n z y[1] y[2] - 1471314862080 n z y[1] y[2] 37 2 5 8 37 2 4 9 - 1471314862080 n z y[1] y[2] - 103229092423945908 n z y[1] y[2] 37 2 3 10 + 60076409927190006 n z y[1] y[2] 37 2 2 11 + 21080467257852996 n z y[1] y[2] 37 11 3 + 6736319563538922 n z y[1] y[2] 37 10 4 37 9 5 - 7402019644010838 n z y[1] y[2] + 1013313314884416 n z y[1] y[2] 37 8 6 37 7 7 - 1383396544000968 n z y[1] y[2] + 1834592591089776 n z y[1] y[2] 37 6 8 37 5 9 - 1383396544000968 n z y[1] y[2] + 1013313314884416 n z y[1] y[2] 37 4 10 - 7402019644010838 n z y[1] y[2] 37 3 11 38 11 2 + 6736319563538922 n z y[1] y[2] + 71443379986020 n z y[1] 38 11 2 38 10 3 + 71443379986020 n z y[2] + 144808559138880 n z y[1] 38 10 2 - 1526348242342560 n z y[1] y[2] 38 10 2 38 10 3 - 1526348242342560 n z y[1] y[2] + 144808559138880 n z y[2] 38 9 4 38 9 2 2 - 357003795217860 n z y[1] - 286568259694500 n z y[1] y[2] 38 9 4 38 8 5 - 357003795217860 n z y[2] - 821312719920 n z y[1] 38 8 4 38 8 4 - 95640305886720 n z y[1] y[2] - 95640305886720 n z y[1] y[2] 38 8 5 38 7 5 - 821312719920 n z y[2] - 9463371760080 n z y[1] y[2] 38 7 5 38 6 6 - 9463371760080 n z y[1] y[2] + 6062166104400 n z y[1] y[2] 38 6 6 38 5 8 + 6062166104400 n z y[1] y[2] - 2842051559760 n z y[1] 38 5 8 38 4 9 - 2842051559760 n z y[2] - 18885289687740 n z y[1] 38 4 8 38 4 8 - 122422418310780 n z y[1] y[2] - 122422418310780 n z y[1] y[2] 38 4 9 38 3 10 - 18885289687740 n z y[2] + 156198078369900 n z y[1] 38 3 10 38 2 11 + 156198078369900 n z y[2] - 35721689993010 n z y[1] 38 2 10 38 2 9 2 + 1025316360257280 n z y[1] y[2] + 143284129847250 n z y[1] y[2] 38 2 2 9 38 2 10 + 143284129847250 n z y[1] y[2] + 1025316360257280 n z y[1] y[2] 38 2 11 38 10 2 - 35721689993010 n z y[2] + 501031882085280 n z y[1] y[2] 38 8 4 38 7 5 + 218062724197500 n z y[1] y[2] + 9463371760080 n z y[1] y[2] 38 6 6 38 5 7 - 12124332208800 n z y[1] y[2] + 9463371760080 n z y[1] y[2] 38 4 8 38 2 10 + 218062724197500 n z y[1] y[2] + 501031882085280 n z y[1] y[2] 38 11 2 38 10 3 - 35721689993010 n y[1] y[2] - 301006637508780 n y[1] y[2] 38 9 4 38 8 5 + 375889084905600 n y[1] y[2] + 3663364279680 n y[1] y[2] 38 5 8 38 4 9 + 3663364279680 n y[1] y[2] + 375889084905600 n y[1] y[2] 38 3 10 38 2 11 - 301006637508780 n y[1] y[2] - 35721689993010 n y[1] y[2] 37 11 3 37 11 2 - 11750956549311840 n z y[1] + 19394425189169142 n z y[1] y[2] 37 11 2 37 11 3 + 19394425189169142 n z y[1] y[2] - 11750956549311840 n z y[2] 37 10 4 + 104152198173641478 n z y[1] 37 10 2 2 + 414592254663797826 n z y[1] y[2] 37 10 4 37 9 5 + 104152198173641478 n z y[2] + 6998184408604536 n z y[1] 37 9 4 - 44208783290243964 n z y[1] y[2] 37 9 3 2 + 34765288911285996 n z y[1] y[2] 37 9 2 3 + 34765288911285996 n z y[1] y[2] 37 9 4 37 9 5 - 44208783290243964 n z y[1] y[2] + 6998184408604536 n z y[2] 37 8 6 37 8 5 - 1038309352575264 n z y[1] + 4519411421160552 n z y[1] y[2] 37 8 5 37 8 6 + 4519411421160552 n z y[1] y[2] - 1038309352575264 n z y[2] 37 7 7 37 7 6 + 970960972519608 n z y[1] - 718688358044496 n z y[1] y[2] 37 7 6 37 7 7 - 718688358044496 n z y[1] y[2] + 970960972519608 n z y[2] 37 6 8 37 6 7 - 429463499126424 n z y[1] + 709147699905456 n z y[1] y[2] 37 6 7 37 6 8 + 709147699905456 n z y[1] y[2] - 429463499126424 n z y[2] 37 5 9 37 5 8 - 5757265984665720 n z y[1] - 3865049167616616 n z y[1] y[2] 37 5 8 37 5 9 - 3865049167616616 n z y[1] y[2] - 5757265984665720 n z y[2] 37 4 10 37 4 9 - 108536742650793576 n z y[1] + 12239555167253736 n z y[1] y[2] 37 4 9 37 4 10 + 12239555167253736 n z y[1] y[2] - 108536742650793576 n z y[2] 37 3 11 37 3 9 2 + 18133319470504602 n z y[1] - 33880397305420596 n z y[1] y[2] 37 3 2 9 37 3 11 - 33880397305420596 n z y[1] y[2] + 18133319470504602 n z y[2] 37 2 11 + 1111467229640397 n z y[1] y[2] 37 2 10 2 - 198074975232223803 n z y[1] y[2] 37 2 2 10 - 198074975232223803 n z y[1] y[2] 37 2 11 + 1111467229640397 n z y[1] y[2] 37 11 2 - 19606893364178799 n z y[1] y[2] 37 9 4 37 8 5 + 30567603114796188 n z y[1] y[2] - 568656952864896 n z y[1] y[2] 37 5 8 37 4 9 - 568656952864896 n z y[1] y[2] + 30567603114796188 n z y[1] y[2] 37 2 11 37 11 3 - 19606893364178799 n z y[1] y[2] - 6736319563538922 n y[1] y[2] 37 10 4 37 9 5 + 7402019644010838 n y[1] y[2] - 1013313314884416 n y[1] y[2] 37 8 6 37 7 7 + 1383396544000968 n y[1] y[2] - 1834592591089776 n y[1] y[2] 37 6 8 37 5 9 + 1383396544000968 n y[1] y[2] - 1013313314884416 n y[1] y[2] 37 4 10 37 3 11 + 7402019644010838 n y[1] y[2] - 6736319563538922 n y[1] y[2] 36 11 4 + 2488893924519483012 n z y[1] 36 11 3 - 670144268856385920 n z y[1] y[2] 36 11 2 2 - 2105386228277192952 n z y[1] y[2] 36 11 3 - 670144268856385920 n z y[1] y[2] 36 11 4 36 10 5 + 2488893924519483012 n z y[2] - 63079151636544912 n z y[1] 36 10 4 + 4995449915616499404 n z y[1] y[2] 36 10 3 2 - 2542209915770754900 n z y[1] y[2] 36 10 2 3 - 2542209915770754900 n z y[1] y[2] 36 10 4 36 10 5 + 4995449915616499404 n z y[1] y[2] - 63079151636544912 n z y[2] 36 9 6 36 9 5 + 45157504647918528 n z y[1] + 368393628699010656 n z y[1] y[2] 36 9 4 2 36 9 3 3 - 200400529545216 n z y[1] y[2] + 889584353280 n z y[1] y[2] 36 9 2 4 - 200400529545216 n z y[1] y[2] 36 9 5 36 9 6 + 368393628699010656 n z y[1] y[2] + 45157504647918528 n z y[2] 36 8 7 36 8 6 - 78921136270583280 n z y[1] + 33931914590902080 n z y[1] y[2] 36 8 5 2 36 8 4 3 - 444792176640 n z y[1] y[2] + 889584353280 n z y[1] y[2] 36 8 3 4 36 8 2 5 + 889584353280 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 8 6 36 8 7 + 33931914590902080 n z y[1] y[2] - 78921136270583280 n z y[2] 36 7 8 36 7 7 + 76169247383948616 n z y[1] - 31810823196797088 n z y[1] y[2] 36 7 6 2 36 7 5 3 - 444792176640 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 7 4 4 36 7 3 5 + 889584353280 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 7 2 6 36 7 7 - 444792176640 n z y[1] y[2] - 31810823196797088 n z y[1] y[2] 36 7 8 36 6 9 + 76169247383948616 n z y[2] - 40932475697310048 n z y[1] 36 6 8 36 6 7 2 + 16640613133032192 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 6 6 3 36 6 5 4 - 444792176640 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 6 4 5 36 6 3 6 - 444792176640 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 6 2 7 36 6 8 - 444792176640 n z y[1] y[2] + 16640613133032192 n z y[1] y[2] 36 6 9 36 5 10 - 40932475697310048 n z y[2] + 54930247602373872 n z y[1] 36 5 9 - 370403186956430448 n z y[1] y[2] 36 5 8 2 36 5 7 3 + 200845321721856 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 5 6 4 36 5 5 5 - 444792176640 n z y[1] y[2] + 889584353280 n z y[1] y[2] 36 5 4 6 36 5 3 7 - 444792176640 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 5 2 8 + 200845321721856 n z y[1] y[2] 36 5 9 36 5 10 - 370403186956430448 n z y[1] y[2] + 54930247602373872 n z y[2] 36 4 11 - 2345830054832771010 n z y[1] 36 4 10 - 3726049710112978602 n z y[1] y[2] 36 4 9 2 36 4 8 3 + 7625961004226819580 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 4 7 4 36 4 6 5 - 444792176640 n z y[1] y[2] + 889584353280 n z y[1] y[2] 36 4 5 6 36 4 4 7 + 889584353280 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 4 3 8 36 4 2 9 - 444792176640 n z y[1] y[2] + 7625961004226819580 n z y[1] y[2] 36 4 10 - 3726049710112978602 n z y[1] y[2] 36 4 11 - 2345830054832771010 n z y[2] 36 3 11 - 141528448308058026 n z y[1] y[2] 36 3 10 2 - 2151261509029931760 n z y[1] y[2] 36 3 9 3 36 3 8 4 - 444792176640 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 3 7 5 36 3 6 6 + 889584353280 n z y[1] y[2] + 889584353280 n z y[1] y[2] 36 3 5 7 36 3 4 8 + 889584353280 n z y[1] y[2] - 444792176640 n z y[1] y[2] 36 3 3 9 - 444792176640 n z y[1] y[2] 36 3 2 10 - 2151261509029931760 n z y[1] y[2] 36 3 11 - 141528448308058026 n z y[1] y[2] 36 2 11 2 + 857774825842754232 n z y[1] y[2] 36 2 10 3 + 4396193978969364522 n z y[1] y[2] 36 2 9 4 - 7422495385204808388 n z y[1] y[2] 36 2 8 5 36 2 7 6 - 195986584958976 n z y[1] y[2] + 889584353280 n z y[1] y[2] 36 2 6 7 36 2 5 8 + 889584353280 n z y[1] y[2] - 195986584958976 n z y[1] y[2] 36 2 4 9 - 7422495385204808388 n z y[1] y[2] 36 2 3 10 + 4396193978969364522 n z y[1] y[2] 36 2 2 11 + 857774825842754232 n z y[1] y[2] 36 11 3 + 781943351610923670 n z y[1] y[2] 36 10 4 - 847941390707285628 n z y[1] y[2] 36 9 5 + 34670542444618320 n z y[1] y[2] 36 8 6 - 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