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%I #15 Aug 14 2020 11:49:17
%S 1,0,2,0,22,33,181,375,1120,2342,5467,10668,21660,39562,72816,125484,
%T 215161,352424,572086,897867,1394315,2110350,3159826,4635480,6731131,
%U 9612072,13595657,18964299,26221103,35828058,48562922,65155439,86777107,114549589,150198041,195400674,252651242
%N Poincaré series for invariant polynomial functions on the space of binary forms of degree 13.
%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=13)
%e The Poincaré series is (1 + t^4 - t^6 + 19t^8 + 31t^10 + 157t^12 + 321t^14 + 885t^16 + 1756t^18 + 3794t^20 + 6856t^22 + 12788t^24 + 21324t^26 + 35633t^28 + 55326t^30 + 85174t^32 + 124064t^34 + 178645t^36 + 246238t^38 + 334814t^40 + 439321t^42 + 568305t^44 + 712862t^46 + 881834t^48 + 1061455t^50 + 1259989t^52 + 1459221t^54 + 1666984t^56 + 1860904t^58 + 2049854t^60 + 2209072t^62 + 2349306t^64 + 2446352t^66 + 2514111t^68 + 2530530t^70 + 2514111t^72 + 2446352t^74 + 2349306t^76 + 2209072t^78 + 2049854t^80 + 1860904t^82 + 1666984t^84 + 1459221t^86 + 1259989t^88 + 1061455t^90 + 881834t^92 + 712862t^94 + 568305t^96 + 439321t^98 + 334814t^100 + 246238t^102 + 178645t^104 + 124064t^106 + 85174t^108 + 55326t^110 + 35633t^112 + 21324t^114 + 12788t^116 + 6856t^118 + 3794t^120 + 1756t^122 + 885t^124 + 321t^126 + 157t^128 + 31t^130 + 19t^132 - t^134 + t^136 + t^140)/ (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)
%p nmax := 120 :
%p (1 + t^4 - t^6 + 19*t^8 + 31*t^10 + 157*t^12 + 321*t^14 + 885*t^16 + 1756*t^18 + 3794*t^20 + 6856*t^22 + 12788*t^24 + 21324*t^26 + 35633*t^28 + 55326*t^30 + 85174*t^32 + 124064*t^34 + 178645*t^36 + 246238*t^38 + 334814*t^40 + 439321*t^42 + 568305*t^44 + 712862*t^46 + 881834*t^48 + 1061455*t^50 + 1259989*t^52 + 1459221*t^54 + 1666984*t^56 + 1860904*t^58 + 2049854*t^60 + 2209072*t^62 + 2349306*t^64 + 2446352*t^66 + 2514111*t^68 + 2530530*t^70 + 2514111*t^72 + 2446352*t^74 + 2349306*t^76 + 2209072*t^78 + 2049854*t^80 + 1860904*t^82 + 1666984*t^84 + 1459221*t^86 + 1259989*t^88 + 1061455*t^90 + 881834*t^92 + 712862*t^94 + 568305*t^96 + 439321*t^98 + 334814*t^100 + 246238*t^102 + 178645*t^104 + 124064*t^106 + 85174*t^108 + 55326*t^110 + 35633*t^112 + 21324*t^114 + 12788*t^116 + 6856*t^118 + 3794*t^120 + 1756*t^122 + 885*t^124 + 321*t^126 + 157*t^128 + 31*t^130 + 19*t^132 - t^134 + t^136 + t^140)/ (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) ;
%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