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Number of ways to place 2 non-attacking wazirs on an n X n toroidal board.
+10
8
0, 2, 18, 88, 250, 558, 1078, 1888, 3078, 4750, 7018, 10008, 13858, 18718, 24750, 32128, 41038, 51678, 64258, 79000, 96138, 115918, 138598, 164448, 193750, 226798, 263898, 305368, 351538, 402750, 459358, 521728, 590238, 665278, 747250, 836568, 933658, 1038958
COMMENTS
A wazir is a leaper [0,1].
FORMULA
a(n) = n^2*(n^2-5)/2, n>=3.
G.f.: 2*x^2 * (2*x^5 - 9*x^4 + 15*x^3 - 9*x^2 - 4*x - 1)/(x-1)^5.
Number of ways to place 3 non-attacking wazirs on an n X n toroidal board.
+10
8
0, 0, 6, 208, 1300, 4908, 14112, 34112, 73008, 142700, 259908, 447312, 734812, 1160908, 1774200, 2635008, 3817112, 5409612, 7518908, 10270800, 13812708, 18316012, 23978512, 31027008, 39720000, 50350508, 63249012, 78786512, 97377708, 119484300, 145618408
COMMENTS
A wazir is a leaper [0,1].
FORMULA
a(n) = n^2*(n^4-15*n^2+62)/6, n>=4.
G.f.: -2*x^3 * (3*x^7 - 15*x^6 + 25*x^5 - 7*x^4 - 17*x^3 - 15*x^2 + 83*x + 3)/(x-1)^7.
Number of ways to place 4 non-attacking wazirs on an n X n toroidal board.
+10
8
0, 0, 0, 228, 3850, 27225, 122892, 423152, 1213380, 3046025, 6907890, 14454972, 28330822, 52586065, 93218400, 158854080, 261593552, 418045617, 650576150, 988799100, 1471339170, 2147897257, 3081651412, 4352027760, 6057877500, 8321097785, 11290735962
FORMULA
a(n) = n^2*(n^2-11)*(n^4 - 19n^2 + 114)/24, n>=5.
G.f.: x^4 * (8x^9 - 54x^8 + 189x^7 - 551x^6 + 1404x^5 - 2552x^4 + 2685x^3 - 783x^2 - 1798x - 228)/(x-1)^9.
MATHEMATICA
CoefficientList[Series[x^3*(8 x^9 - 54 x^8 + 189 x^7 - 551 x^6 + 1404 x^5 - 2552 x^4 + 2685 x^3 - 783 x^2 - 1798 x - 228)/(x - 1)^9, {x, 0, 20}], x] (* Wesley Ivan Hurt, Jan 19 2017 *)
Number of ways to place 5 non-attacking wazirs on an n X n toroidal board.
+10
8
0, 0, 0, 128, 6745, 100332, 754453, 3830016, 15038541, 49207020, 140410699, 360001152, 846775007, 1855033964, 3828109545, 7507096576, 14087087961, 25436160108, 44395753647, 75184958080, 123935571963, 199389702380, 313797119069, 484055619840, 733144325125
FORMULA
a(n) = n^2*(n^8 - 50n^6 + 995n^4 - 9370n^2 + 35424)/120, n>=6.
G.f.: -x^4 * (10x^12 - 110x^11 + 685x^10 - 2771x^9 + 6946x^8 - 9350x^7 + 1710x^6 + 15214x^5 - 21392x^4 + 656x^3 + 33177x^2 + 5337x + 128)/(x-1)^11.
Number of ways to place 6 non-attacking wazirs on an n X n toroidal board.
+10
8
0, 0, 0, 56, 7100, 252792, 3378942, 26249184, 144455454, 625745100, 2271361422, 7192874328, 20427662398, 53065637212, 127956238350, 289628321664, 620834113614, 1269178026012, 2488676915070, 4702895069400, 8598589878606, 15261688799500, 26371002575326
FORMULA
a(n) = n^2*(n^10 - 75*n^8 + 2365*n^6 - 39285*n^4 + 345034*n^2 - 1288680)/720, n>=7.
G.f.: 2*x^4 * (6*x^15 - 103*x^14 + 873*x^13 - 4241*x^12 + 12757*x^11 - 26112*x^10 + 45344*x^9 - 90774*x^8 + 189180*x^7 - 293907*x^6 + 260273*x^5 - 25077*x^4 - 315215*x^3 - 82430*x^2 - 3186*x - 28)/(x-1)^13.
Number of ways to place 8 non-attacking wazirs on an n X n toroidal board.
+10
8
0, 0, 0, 2, 1550, 546516, 28482279, 585632520, 6829066665, 54504255500, 331490619174, 1642426038486, 6930083422496, 25686190415144, 85541928717375, 260349711114720, 733731834393719, 1934755847570808, 4813391235753128, 11375736647373750, 25684539545337246
FORMULA
a(n) = n^2*(n^14 - 140n^12 + 8722n^10 - 313880n^8 + 7061089n^6 - 99573740n^4 + 817978188n^2 - 3033601200)/40320, n>=9.
G.f.: x^4 * (16x^21 - 566x^20 + 8182x^19 - 67700x^18 + 377824x^17 - 1531112x^16 + 4601788x^15 - 10205035x^14 + 16637339x^13 - 21628151x^12 + 32135719x^11 - 68863352x^10 + 138461546x^9 - 189569712x^8 + 133644570x^7 + 20663373x^6 - 378949513x^5 - 174710713x^4 - 19400947x^3 - 520438x^2 - 1516x - 2)/(x-1)^17.
Number of ways to place 9 nonattacking wazirs on an n X n toroidal board.
+10
1
0, 0, 0, 0, 250, 480916, 54916456, 1962132800, 34690541994, 385983794500, 3095143575007, 19437996015280, 100963195651565, 450398154002132, 1773257833600750, 6288010190509312, 20398342362118678, 61282868654684052, 172190699515632837, 456120623076014000
FORMULA
Explicit formula: n^18/362880 - n^16/2016 + 349*n^14/8640 - 467*n^12/240 + 1049629*n^10/17280 - 121049*n^8/96 + 1546301783*n^6/90720 - 346878319*n^4/2520 + 4595485*n^2/9, n>=10.
G.f.: -x^5*(18*x^23 - 854*x^22 + 15942*x^21 - 168082*x^20 + 1174353*x^19 - 5878707*x^18 + 22139332*x^17 - 65539648*x^16 + 159915785*x^15 - 334575275*x^14 + 598795512*x^13 - 842713520*x^12 + 703597341*x^11 + 289921121*x^10 - 2021527454*x^9 + 3166171570*x^8 - 1944444195*x^7 - 501647511*x^6 + 11035282966*x^5 + 6335694166*x^4 + 1000714522*x^3 + 45821802*x^2 + 476166*x + 250)/(x-1)^19.
Number of ways to place 10 nonattacking wazirs on an n X n toroidal board.
+10
1
0, 0, 0, 0, 10, 308574, 81442802, 5296005568, 146127335256, 2309813476870, 24738873315596, 198759048859008, 1279605298916568, 6906427308782106, 32277449304595350, 133788325435448576, 500896430870051174, 1718268150463137018, 5462521782760829320, 16243031089247644800
FORMULA
Explicit formula: n^20/3628800 - n^18/16128 + 773*n^16/120960 - 761*n^14/1920 + 2820613*n^12/172800 - 356093*n^10/768 + 412940467*n^8/45360 - 2408161207*n^6/20160 + 24029851729*n^4/25200 - 3541971*n^2, n>=11.
G.f.: 2*x^5*(10*x^26 - 615*x^25 + 14637*x^24 - 193410*x^23 + 1669110*x^22 - 10270682*x^21 + 47718030*x^20 - 174153546*x^19 + 511148331*x^18 - 1213451007*x^17 + 2302816572*x^16 - 3418379599*x^15 + 4006461091*x^14 - 4626995415*x^13 + 8410419611*x^12 - 19068629603*x^11 + 33871890471*x^10 - 39181017568*x^9 + 18018811352*x^8 - 5120263515*x^7 - 178499919965*x^6 - 123414145507*x^5 - 25801931589*x^4 - 1825246983*x^3 - 37482424*x^2 - 154182*x - 5)/(x-1)^21.
MATHEMATICA
CoefficientList[Series[2 x^4 (10 x^26 - 615 x^25 + 14637 x^24 - 193410 x^23 + 1669110 x^22 - 10270682 x^21 + 47718030 x^20 - 174153546 x^19 + 511148331 x^18 - 1213451007 x^17 + 2302816572 x^16 - 3418379599 x^15 + 4006461091 x^14 - 4626995415 x^13 + 8410419611 x^12 - 19068629603 x^11 + 33871890471 x^10 - 39181017568 x^9 + 18018811352 x^8 - 5120263515 x^7 - 178499919965 x^6 - 123414145507 x^5 - 25801931589 x^4 - 1825246983 x^3 - 37482424 x^2-154182 x - 5) / (x - 1)^21, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 04 2013 *)
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