A083024 Molien series for action of SL(3,C) on ternary forms of degree 4.

1, 1, 2, 4, 7, 11, 19, 29, 44, 67, 98, 139, 199, 275, 375, 509, 678, 890, 1165, 1501, 1916, 2431, 3053, 3801, 4711, 5788, 7063, 8580, 10353, 12420, 14841, 17633, 20850, 24565, 28807, 33641, 39161, 45404, 52455, 60427, 69372, 79392, 90627, 103143, 117065, 132561

Offset: 0

Author

Benoit Cloitre, Jun 01 2003

Keywords

• nonn
• easy

Comments

These are the coefficients of the expansion in powers of z^4, the other coefficients being zero.

References

• J-M. Kantor, Où en sont les mathématiques. La formule de Molien-Weyl, SMF, Vuibert, p. 79

Links

• T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.
• Index entries for Molien series.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,2,0,0,1,0,-1,0,1,0,-1,0,0,-2,0,2,0,1,0,0,-1,-1,1).

Crossrefs

Cf. A008615.

Programs

Maple

seq(coeff(series( (1 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 3*x^11 + 4*x^12 + 4*x^13 + 3*x^14 + 4*x^15 + 3*x^16 + 2*x^17 + 3*x^18 + 2*x^19 + x^20 + x^21 + x^25)/(1 - x^1 - x^2 + x^5 + 2*x^7 - 2*x^9 - x^12 + x^14 - x^16 + x^18 + 2*x^21 - 2*x^23 - x^25 + x^28 + x^29 - x^30), x, n+1), x, n), n = 0..45); # Georg Fischer, Jan 24 2021

PARI

a(n)=polcoeff((1+z^9+z^12+z^15+2*z^18+3*z^21+2*z^24+3*z^27+4*z^30+3*z^33 +4*z^36+4*z^39+3*z^42+4*z^45+3*z^48+2*z^51+3*z^54+2*z^57+z^60+z^63+z^75) /(1-z^3)/(1-z^6)/(1-z^9)/(1-z^12)/(1-z^15)/(1-z^18)/(1- z^27)+O(z^(n+1)),n)

Formula

G.f.: (1 + z^9 + z^12 + z^15 + 2*z^18 + 3*z^21 + 2*z^24 + 3*z^27 + 4*z^30 + 3*z^33 + 4*z^36 + 4*z^39 + 3*z^42 + 4*z^45 + 3*z^48 + 2*z^51 + 3*z^54 + 2*z^57 + z^60 + z^63 + z^75)/(1-z^3)/(1-z^6)/(1-z^9)/(1-z^12)/(1-z^15)/(1-z^18)/(1-z^27).