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%I #13 Aug 14 2020 11:48:31
%S 1,0,1,1,3,4,13,18,47,84,177,320,639,1120,2077,3581,6235,10395,17344,
%T 27940,44848,70180,108921,165817,250256,371558,546960,794363,1143783,
%U 1628190,2299144,3214042,4459495,6133164,8375820,11349269,15278595,20423345,27136816,35827488,47037493,61397294,79726515,102977471,132370606,169322488,215620140,273339320,345063648,433787088,543198659,677563207,842079818,1042751237,1286826668,1582652314,1940231900,2371051392,2888771603,3509044867,4250358055,5133832789,6184270777,7429930460
%N Poincaré series for invariant polynomial functions on the space of binary forms of degree 16.
%C Many of these Poincaré series has every other term zero, in which case these zeros have been omitted.
%H Andries Brouwer, <a href="http://www.win.tue.nl/~aeb/math/poincare.html">Poincaré Series</a> (See n=16)
%e The Poincaré series is (1 + t^4 + 2t^5 + 8t^6 + 11t^7 + 28t^8 + 51t^9 + 102t^10 + 177t^11 + 340t^12 + 561t^13 + 980t^14 + 1586t^15 + 2565t^16 + 3955t^17 + 6095t^18 + 8991t^19 + 13206t^20 + 18815t^21 + 26498t^22 + 36437t^23 + 49596t^24 + 66028t^25 + 87003t^26 + 112578t^27 + 144034t^28 + 181363t^29 + 226014t^30 + 277437t^31 + 337179t^32 + 404317t^33 + 479951t^34 + 562691t^35 + 653453t^36 + 749807t^37 + 852481t^38 + 958443t^39 + 1067723t^40 + 1176799t^41 + 1285637t^42 + 1389850t^43 + 1489589t^44 + 1580460t^45 + 1662409t^46 + 1731403t^47 + 1788102t^48 + 1828489t^49 + 1854175t^50 + 1862064t^51 + 1854175t^52 + 1828489t^53 + 1788102t^54 + 1731403t^55 + 1662409t^56 + 1580460t^57 + 1489589t^58 + 1389850t^59 + 1285637t^60 + 1176799t^61 + 1067723t^62 + 958443t^63 + 852481t^64 + 749807t^65 + 653453t^66 + 562691t^67 + 479951t^68 + 404317t^69 + 337179t^70 + 277437t^71 + 226014t^72 + 181363t^73 + 144034t^74 + 112578t^75 + 87003t^76 + 66028t^77 + 49596t^78 + 36437t^79 + 26498t^80 + 18815t^81 + 13206t^82 + 8991t^83 + 6095t^84 + 3955t^85 + 2565t^86 + 1586t^87 + 980t^88 + 561t^89 + 340t^90 + 177t^91 + 102t^92 + 51t^93 + 28t^94 + 11t^95 + 8t^96 + 2t^97 + t^98 + t^102) / (1 - t^2)(1 - t^3)(1 - t^4)(1 - t^5)(1 - t^6)(1 - t^7) (1 - t^8)(1 - t^9)(1 - t^10)(1 - t^11)(1 - t^12)(1 - t^13)(1 - t^14) (1 - t^15)
%p nmax := 120 :
%p (1 + t^4 + 2*t^5 + 8*t^6 + 11*t^7 + 28*t^8 + 51*t^9 + 102*t^10 + 177*t^11 + 340*t^12 + 561*t^13 + 980*t^14 + 1586*t^15 + 2565*t^16 + 3955*t^17 + 6095*t^18 + 8991*t^19 + 13206*t^20 + 18815*t^21 + 26498*t^22 + 36437*t^23 + 49596*t^24 + 66028*t^25 + 87003*t^26 + 112578*t^27 + 144034*t^28 + 181363*t^29 + 226014*t^30 + 277437*t^31 + 337179*t^32 + 404317*t^33 + 479951*t^34 + 562691*t^35 + 653453*t^36 + 749807*t^37 + 852481*t^38 + 958443*t^39 + 1067723*t^40 + 1176799*t^41 + 1285637*t^42 + 1389850*t^43 + 1489589*t^44 + 1580460*t^45 + 1662409*t^46 + 1731403*t^47 + 1788102*t^48 + 1828489*t^49 + 1854175*t^50 + 1862064*t^51 + 1854175*t^52 + 1828489*t^53 + 1788102*t^54 + 1731403*t^55 + 1662409*t^56 + 1580460*t^57 + 1489589*t^58 + 1389850*t^59 + 1285637*t^60 + 1176799*t^61 + 1067723*t^62 + 958443*t^63 + 852481*t^64 + 749807*t^65 + 653453*t^66 + 562691*t^67 + 479951*t^68 + 404317*t^69 + 337179*t^70 + 277437*t^71 + 226014*t^72 + 181363*t^73 + 144034*t^74 + 112578*t^75 + 87003*t^76 + 66028*t^77 + 49596*t^78 + 36437*t^79 + 26498*t^80 + 18815*t^81 + 13206*t^82 + 8991*t^83 + 6095*t^84 + 3955*t^85 + 2565*t^86 + 1586*t^87 + 980*t^88 + 561*t^89 + 340*t^90 + 177*t^91 + 102*t^92 + 51*t^93 + 28*t^94 + 11*t^95 + 8*t^96 + 2*t^97 + t^98 + t^102) / (1 - t^2)/(1 - t^3)/(1 - t^4)/(1 - t^5)/(1 - t^6)/(1 - t^7) /(1 - t^8)/(1 - t^9)/(1 - t^10)/(1 - t^11)/(1 - t^12)/(1 - t^13)/(1 - t^14) /(1 - t^15) ;
%p taylor(%,t=0,nmax) ;
%p gfun[seriestolist](%) ; # _R. J. Mathar_, Oct 26 2017
%Y For these Poincaré series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.
%K nonn
%O 0,5
%A _N. J. A. Sloane_, Oct 20 2017
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