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%I #14 Aug 14 2020 11:48:45
%S 1,0,3,1,36,80,418,1111,3581,8899,22786,51286,114049,234754,472443,
%T 902625,1683916,3024451,5313062,9063638,15158162,24760532,39743317,
%U 62563090,96977396,147874275,222433862,329908935,483445738,699822112,1002221943,1419949064,1992553143
%N Poincaré series for invariant polynomial functions on the space of binary forms of degree 15.
%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=915)
%e The Poincaré series is (1 + 2t^4 + 32t^8 + 76t^10 + 378t^12 + 995t^14 + 3048t^16 + 7294t^18 + 17681t^20 + 37736t^22 + 78903t^24 + 152321t^26 + 285968t^28 + 507762t^30 + 876759t^32 + 1451423t^34 + 2341739t^36 + 3653241t^38 + 5568497t^40 + 8254649t^42 + 11983447t^44 + 16987847t^46 + 23631274t^48 + 32196429t^50 + 43116834t^52 + 56681420t^54 + 73342055t^56 + 93320393t^58 + 117007543t^60 + 144461993t^62 + 175919353t^64 + 211175615t^66 + 250222591t^68 + 292516508t^70 + 337751801t^72 + 385016863t^74 + 433713649t^76 + 482605505t^78 + 530877973t^80 + 577086324t^82 + 620343376t^84 + 659172312t^86 + 692798202t^88 + 719914717t^90 + 740045690t^92 + 752239053t^94 + 756462172t^96 + 752239053t^98 + 740045690t^100 + 719914717t^102 + 692798202t^104 + 659172312t^106 + 620343376t^108 + 577086324t^110 + 530877973t^112 + 482605505t^114 + 433713649t^116 + 385016863t^118 + 337751801t^120 + 292516508t^122 + 250222591t^124 + 211175615t^126 + 175919353t^128 + 144461993t^130 + 117007543t^132 + 93320393t^134 + 73342055t^136 + 56681420t^138 + 43116834t^140 + 32196429t^142 + 23631274t^144 + 16987847t^146 + 11983447t^148 + 8254649t^150 + 5568497t^152 + 3653241t^154 + 2341739t^156 + 1451423t^158 + 876759t^160 + 507762t^162 + 285968t^164 + 152321t^166 + 78903t^168 + 37736t^170 + 17681t^172 + 7294t^174 + 3048t^176 + 995t^178 + 378t^180 + 76t^182 + 32t^184 + 2t^188 + t^192) / (1 - t^4)(1 - t^6)(1 - t^8)(1 - t^10)(1 - t^12)(1 - t^14)(1 - t^16) (1 - t^18)(1 - t^20)(1 - t^22)(1 - t^24)(1 - t^26)(1 - t^28)
%p nmax := 120 :
%p (1 + 2*t^4 + 32*t^8 + 76*t^10 + 378*t^12 + 995*t^14 + 3048*t^16 + 7294*t^18 + 17681*t^20 + 37736*t^22 + 78903*t^24 + 152321*t^26 + 285968*t^28 + 507762*t^30 + 876759*t^32 + 1451423*t^34 + 2341739*t^36 + 3653241*t^38 + 5568497*t^40 + 8254649*t^42 + 11983447*t^44 + 16987847*t^46 + 23631274*t^48 + 32196429*t^50 + 43116834*t^52 + 56681420*t^54 + 73342055*t^56 + 93320393*t^58 + 117007543*t^60 + 144461993*t^62 + 175919353*t^64 + 211175615*t^66 + 250222591*t^68 + 292516508*t^70 + 337751801*t^72 + 385016863*t^74 + 433713649*t^76 + 482605505*t^78 + 530877973*t^80 + 577086324*t^82 + 620343376*t^84 + 659172312*t^86 + 692798202*t^88 + 719914717*t^90 + 740045690*t^92 + 752239053*t^94 + 756462172*t^96 + 752239053*t^98 + 740045690*t^100 + 719914717*t^102 + 692798202*t^104 + 659172312*t^106 + 620343376*t^108 + 577086324*t^110 + 530877973*t^112 + 482605505*t^114 + 433713649*t^116 + 385016863*t^118 + 337751801*t^120 + 292516508*t^122 + 250222591*t^124 + 211175615*t^126 + 175919353*t^128 + 144461993*t^130 + 117007543*t^132 + 93320393*t^134 + 73342055*t^136 + 56681420*t^138 + 43116834*t^140 + 32196429*t^142 + 23631274*t^144 + 16987847*t^146 + 11983447*t^148 + 8254649*t^150 + 5568497*t^152 + 3653241*t^154 + 2341739*t^156 + 1451423*t^158 + 876759*t^160 + 507762*t^162 + 285968*t^164 + 152321*t^166 + 78903*t^168 + 37736*t^170 + 17681*t^172 + 7294*t^174 + 3048*t^176 + 995*t^178 + 378*t^180 + 76*t^182 + 32*t^184 + 2*t^188 + t^192) / (1 - t^4)/(1 - t^6)/(1 - t^8)/(1 - t^10)/(1 - t^12)/(1 - t^14)/(1 - t^16) /(1 - t^18)/(1 - t^20)/(1 - t^22)/(1 - t^24)/(1 - t^26)/(1 - t^28) ;
%p taylor(%,t=0,nmax) ;
%p gfun[seriestolist](%) ;
%p seq( %[1+2*i],i=0..nmax/2-1) ; # _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 0,3
%A _N. J. A. Sloane_, Oct 20 2017
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