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%I #14 Aug 14 2020 11:48:20
%S 1,0,1,1,4,5,20,35,102,217,540,1160,2634,5467,11463,22786,44848,85068,
%T 159018,288914,516643,903256,1554696,2626217,4372347,7163317,11580760,
%U 18462388,29078307,45236642,69602057,105917976,159571937,238035458,351841043,515413775,748727920
%N Poincaré series for invariant polynomial functions on the space of binary forms of degree 20.
%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=20)
%e The Poincaré series is (1 + 2t^4 + 3t^5 + 14t^6 + 26t^7 + 74t^8 + 159t^9 + 386t^10 + 813t^11 + 1786t^12 + 3581t^13 + 7194t^14 + 13690t^15 + 25662t^16 + 46264t^17 + 81972t^18 + 140858t^19 + 237716t^20 + 391489t^21 + 633566t^22 + 1004435t^23 + 1567003t^24 + 2401414t^25 + 3626076t^26 + 5390337t^27 + 7904749t^28 + 11431403t^29 + 16326733t^30 + 23026390t^31 + 32104634t^32 + 44251748t^33 + 60350746t^34 + 81444897t^35 + 108834679t^36 + 144027146t^37 + 188856601t^38 + 245409166t^39 + 316164054t^40 + 403886629t^41 + 511790842t^42 + 643385302t^43 + 802659024t^44 + 993869808t^45 + 1221746711t^46 + 1491212905t^47 + 1807606172t^48 + 2176318945t^49 + 2603044019t^50 + 3093325449t^51 + 3652826842t^52 + 4286795701t^53 + 5000365547t^54 + 5797926021t^55 + 6683480280t^56 + 7659930019t^57 + 8729496251t^58 + 9892940351t^59 + 11150071817t^60 + 12498910419t^61 + 13936313872t^62 + 15457101887t^63 + 17054812157t^64 + 18720815924t^65 + 20445221332t^66 + 22215981071t^67 + 24019965060t^68 + 25842070888t^69 + 27666450024t^70 + 29475635493t^71 + 31251911254t^72 + 32976432897t^73 + 34630731555t^74 + 36195812801t^75 + 37653744237t^76 + 38986726195t^77 + 40178723548t^78 + 41214465019t^79 + 42081091938t^80 + 42767062988t^81 + 43263759348t^82 + 43564293507t^83 + 43665034627t^84 + 43564293507t^85 + 43263759348t^86 + 42767062988t^87 + 42081091938t^88 + 41214465019t^89 + 40178723548t^90 + 38986726195t^91 + 37653744237t^92 + 36195812801t^93 + 34630731555t^94 + 32976432897t^95 + 31251911254t^96 + 29475635493t^97 + 27666450024t^98 + 25842070888t^99 + 24019965060t^100 + 22215981071t^101 + 20445221332t^102 + 18720815924t^103 + 17054812157t^104 + 15457101887t^105 + 13936313872t^106 + 12498910419t^107 + 11150071817t^108 + 9892940351t^109 + 8729496251t^110 + 7659930019t^111 + 6683480280t^112 + 5797926021t^113 + 5000365547t^114 + 4286795701t^115 + 3652826842t^116 + 3093325449t^117 + 2603044019t^118 + 2176318945t^119 + 1807606172t^120 + 1491212905t^121 + 1221746711t^122 + 993869808t^123 + 802659024t^124 + 643385302t^125 + 511790842t^126 + 403886629t^127 + 316164054t^128 + 245409166t^129 + 188856601t^130 + 144027146t^131 + 108834679t^132 + 81444897t^133 + 60350746t^134 + 44251748t^135 + 32104634t^136 + 23026390t^137 + 16326733t^138 + 11431403t^139 + 7904749t^140 + 5390337t^141 + 3626076t^142 + 2401414t^143 + 1567003t^144 + 1004435t^145 + 633566t^146 + 391489t^147 + 237716t^148 + 140858t^149 + 81972t^150 + 46264t^151 + 25662t^152 + 13690t^153 + 7194t^154 + 3581t^155 + 1786t^156 + 813t^157 + 386t^158 + 159t^159 + 74t^160 + 26t^161 + 14t^162 + 3t^163 + 2t^164 + t^168) / (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)(1 - t^16)(1 - t^17)(1 - t^18) (1 - t^19)
%p nmax := 120 :
%p (1 + 2*t^4 + 3*t^5 + 14*t^6 + 26*t^7 + 74*t^8 + 159*t^9 + 386*t^10 + 813*t^11 + 1786*t^12 + 3581*t^13 + 7194*t^14 + 13690*t^15 + 25662*t^16 + 46264*t^17 + 81972*t^18 + 140858*t^19 + 237716*t^20 + 391489*t^21 + 633566*t^22 + 1004435*t^23 + 1567003*t^24 + 2401414*t^25 + 3626076*t^26 + 5390337*t^27 + 7904749*t^28 + 11431403*t^29 + 16326733*t^30 + 23026390*t^31 + 32104634*t^32 + 44251748*t^33 + 60350746*t^34 + 81444897*t^35 + 108834679*t^36 + 144027146*t^37 + 188856601*t^38 + 245409166*t^39 + 316164054*t^40 + 403886629*t^41 + 511790842*t^42 + 643385302*t^43 + 802659024*t^44 + 993869808*t^45 + 1221746711*t^46 + 1491212905*t^47 + 1807606172*t^48 + 2176318945*t^49 + 2603044019*t^50 + 3093325449*t^51 + 3652826842*t^52 + 4286795701*t^53 + 5000365547*t^54 + 5797926021*t^55 + 6683480280*t^56 + 7659930019*t^57 + 8729496251*t^58 + 9892940351*t^59 + 11150071817*t^60 + 12498910419*t^61 + 13936313872*t^62 + 15457101887*t^63 + 17054812157*t^64 + 18720815924*t^65 + 20445221332*t^66 + 22215981071*t^67 + 24019965060*t^68 + 25842070888*t^69 + 27666450024*t^70 + 29475635493*t^71 + 31251911254*t^72 + 32976432897*t^73 + 34630731555*t^74 + 36195812801*t^75 + 37653744237*t^76 + 38986726195*t^77 + 40178723548*t^78 + 41214465019*t^79 + 42081091938*t^80 + 42767062988*t^81 + 43263759348*t^82 + 43564293507*t^83 + 43665034627*t^84 + 43564293507*t^85 + 43263759348*t^86 + 42767062988*t^87 + 42081091938*t^88 + 41214465019*t^89 + 40178723548*t^90 + 38986726195*t^91 + 37653744237*t^92 + 36195812801*t^93 + 34630731555*t^94 + 32976432897*t^95 + 31251911254*t^96 + 29475635493*t^97 + 27666450024*t^98 + 25842070888*t^99 + 24019965060*t^100 + 22215981071*t^101 + 20445221332*t^102 + 18720815924*t^103 + 17054812157*t^104 + 15457101887*t^105 + 13936313872*t^106 + 12498910419*t^107 + 11150071817*t^108 + 9892940351*t^109 + 8729496251*t^110 + 7659930019*t^111 + 6683480280*t^112 + 5797926021*t^113 + 5000365547*t^114 + 4286795701*t^115 + 3652826842*t^116 + 3093325449*t^117 + 2603044019*t^118 + 2176318945*t^119 + 1807606172*t^120 + 1491212905*t^121 + 1221746711*t^122 + 993869808*t^123 + 802659024*t^124 + 643385302*t^125 + 511790842*t^126 + 403886629*t^127 + 316164054*t^128 + 245409166*t^129 + 188856601*t^130 + 144027146*t^131 + 108834679*t^132 + 81444897*t^133 + 60350746*t^134 + 44251748*t^135 + 32104634*t^136 + 23026390*t^137 + 16326733*t^138 + 11431403*t^139 + 7904749*t^140 + 5390337*t^141 + 3626076*t^142 + 2401414*t^143 + 1567003*t^144 + 1004435*t^145 + 633566*t^146 + 391489*t^147 + 237716*t^148 + 140858*t^149 + 81972*t^150 + 46264*t^151 + 25662*t^152 + 13690*t^153 + 7194*t^154 + 3581*t^155 + 1786*t^156 + 813*t^157 + 386*t^158 + 159*t^159 + 74*t^160 + 26*t^161 + 14*t^162 + 3*t^163 + 2*t^164 + t^168) / (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)/(1 - t^16)/(1 - t^17)/(1 - t^18) /(1 - t^19) ;
%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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