A246984 Coefficients in Molien series for a 25-dimensional representation of SO(3) wreath Z_2.

1, 0, 1, 1, 4, 5, 14, 20, 52, 90, 201, 371, 792, 1485, 2977, 5557, 10620, 19354, 35467, 62833, 110908, 190860, 325794, 545070, 903160, 1471723, 2374077, 3775806, 5945620, 9247524, 14246994, 21711107, 32790785, 49043622, 72740759, 106941223

Offset

0,5

Links

Table of n, a(n) for n=0..35.

D. R. J. Chillingworth, R. Lauterbach and S. S. Turzi, Molien series and low-degree invariants for a natural action of SO(3) wreath Z_2, arXiv 1407.6738, 2014.

Formula

Chillingworth et al. (2014) (see Eqs. 104-111 and Appendix A) give an explicit g.f.

Maple

Q0 := (1-t^2) *(1-t^3) *(1-t^4)^3 *(1-t^5) *(1-t^6)^2 *(1-t^7)^2
*(1-t^8)^2 *(1-t^9)^2 *(1-t^10)^2 *(1-t^11) *(1-t^12) *(1-t^13) ;
P1 := 1 -t^4 +3*t^5 +3*t^6 +4*t^7 +10*t^8 +12*t^9 +17*t^10 +25*t^11
+30*t^12 +36*t^13 +41*t^14 +41*t^15 +40*t^16 +36*t^17 +16*t^18 -9*t^19
-32*t^20 -74*t^21 -122*t^22 -168*t^23 -223*t^24 -266*t^25 -298*t^26 -
324*t^27 -312*t^28 -274*t^29 -216*t^30 -108*t^31 +30*t^32
+183*t^33 +364*t^34 +546*t^35 +717*t^36 +871*t^37 +961*t^38 +999*t^39 +979*t^40
+859*t^41 +670*t^42 +413*t^43 +83*t^44 -268*t^45 -639*t^46 -1002*t^47 -
1299*t^48 -1536*t^49 -1683*t^50 -1695*t^51 -1601*t^52 -1398*t^53 -1072*t^54
-680*t^55 -238*t^56 +238*t^57 +680*t^58 +1072*t^59 +1398*t^60 +1601*t^61 +1695*t^62
+1683*t^63 +1536*t^64 +1299*t^65 +1002*t^66 +639*t^67 +268*t^68 -83*t^69
-413*t^70 -670*t^71 -859*t^72 -979*t^73 -999*t^74 -961*t^75 -871*t^76
-717*t^77 -546*t^78 -364*t^79 -183*t^80 -30*t^81 +108*t^82 +216*t^83 +274*t^84
+312*t^85 +324*t^86 +298*t^87 +266*t^88 +223*t^89 +168*t^90 +122*t^91 +74*t^92
+32*t^93 +9*t^94 -16*t^95 -36*t^96 -40*t^97 -41*t^98 -41*t^99 -36*t^100
-30*t^101 -25*t^102 -17*t^103 -12*t^104 -10*t^105 -4*t^106 -3*t^107
-3*t^108 +t^109 -t^113 ;
Q1 := Q0 ;
P0 := 1 +t^4 +3*t^5 +11*t^6 +16*t^7 +42*t^8 +80*t^9 +185*t^10 +357*t^11
+752*t^12 +1412*t^13 +2723*t^14 +4937*t^15 +8888*t^16 +15342*t^17
+26146*t^18 +43083*t^19 +69884*t^20 +110398*t^21 +171406*t^22 +260288*t^23
+388723*t^24 +569210*t^25 +820356*t^26 +1161726*t^27 +1620330*t^28
+2224150*t^29 +3009500*t^30
+4012238*t^31 +5276926*t^32 +6845013*t^33 +8764870*t^34 +11078260*t^35
+13830477*t^36 +17054459*t^37 +20782913*t^38 +25029615*t^39
+29802829*t^40 +35086893*t^41 +40855850*t^42 +47055721*t^43 +53620919*t^44
+60456820*t^45 +67458001*t^46 +74494882*t^47 +81431353*t^48 +88115150*t^49
+94396925*t^50 +100121953*t^51 +105148447*t^52 +109343460*t^53 +112595858*t^54
+114815204*t^55 +115941062*t^56 +115941062*t^57 +114815204*t^58 +112595858*t^59
+109343460*t^60 +105148447*t^61 +100121953*t^62 +94396925*t^63 +88115150*t^64
+81431353*t^65 +74494882*t^66 +67458001*t^67 +60456820*t^68 +53620919*t^69
+47055721*t^70 +40855850*t^71 +35086893*t^72 +29802829*t^73 +25029615*t^74
+20782913*t^75 +17054459*t^76 +13830477*t^77 +11078260*t^78 +8764870*t^79
+6845013*t^80 +5276926*t^81 +4012238*t^82 +3009500*t^83 +2224150*t^84
+1620330*t^85 +1161726*t^86 +820356*t^87 +569210*t^88 +388723*t^89 +260288*t^90
+171406*t^91 +110398*t^92 +69884*t^93 +43083*t^94 +26146*t^95 +15342*t^96
+8888*t^97 +4937*t^98 +2723*t^99 +1412*t^100 +752*t^101 +357*t^102 +185*t^103
+80*t^104 +42*t^105 +16*t^106 +11*t^107 +3*t^108 +t^109 +t^113 ;
gfPGamma12 := P1/Q1 ;
gfPGamma02 := P0/Q0 ;
series((gfPGamma12+gfPGamma02)/2, t=0, 100) ;
gfun[seriestolist](%) ; # R. J. Mathar, Oct 25 2014

Crossrefs

Sequence in context: A050164 A188021 A191790 * A306897 A348889 A006904

Adjacent sequences: A246981 A246982 A246983 * A246985 A246986 A246987

Keyword

nonn

Author

N. J. A. Sloane, Sep 15 2014

Status

approved