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%I #17 Aug 14 2020 11:49:34
%S 1,0,1,1,3,3,8,10,20,28,52,73,127,181,291,418,639,902,1330,1848,2634,
%T 3603,4998,6718,9113,12058,16027,20903,27307,35123,45198,57412,72874,
%U 91519,114762,142605,176883,217679,267324,326073,396837,480074,579460,695704,833361,993548
%N Poincaré series for invariant polynomial functions on the space of binary forms of degree 12.
%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=12)
%e The Poincaré series is (1 + t^4 + t^5 + 3t^6 + 4t^7 + 7t^8 + 9t^9 + 17t^10 + 21t^11 + 36t^12 + 45t^13 + 65t^14 + 81t^15 + 110t^16 + 131t^17 + 168t^18 + 193t^19 + 232t^20 + 256t^21 + 293t^22 + 307t^23 + 336t^24 + 339t^25 + 351t^26 + 339t^27 + 336t^28 + 307t^29 + 293t^30 + 256t^31 + 232t^32 + 193t^33 + 168t^34 + 131t^35 + 110t^36 + 81t^37 + 65t^38 + 45t^39 + 36t^40 + 21t^41 + 17t^42 + 9t^43 + 7t^44 + 4t^45 + 3t^46 + t^47 + t^48 + t^52) / (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)
%p nmax := 120 :
%p (1 + t^4 + t^5 + 3*t^6 + 4*t^7 + 7*t^8 + 9*t^9 + 17*t^10 + 21*t^11 + 36*t^12 + 45*t^13 + 65*t^14 + 81*t^15 + 110*t^16 + 131*t^17 + 168*t^18 + 193*t^19 + 232*t^20 + 256*t^21 + 293*t^22 + 307*t^23 + 336*t^24 + 339*t^25 + 351*t^26 + 339*t^27 + 336*t^28 + 307*t^29 + 293*t^30 + 256*t^31 + 232*t^32 + 193*t^33 + 168*t^34 + 131*t^35 + 110*t^36 + 81*t^37 + 65*t^38 + 45*t^39 + 36*t^40 + 21*t^41 + 17*t^42 + 9*t^43 + 7*t^44 + 4*t^45 + 3*t^46 + t^47 + t^48 + t^52) / (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) ;
%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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