A089597 G.f.: (1+x)*(1+x^3)*(1+x^5)*(1+x^7)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)).

1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 14, 18, 23, 27, 31, 38, 46, 52, 59, 69, 80, 90, 100, 114, 130, 143, 157, 176, 196, 214, 233, 257, 283, 306, 330, 360, 392, 421, 451, 488, 527, 562, 599, 643, 689, 732, 776, 828, 883, 933, 985, 1046, 1109, 1168, 1229, 1299, 1372, 1440, 1510

Offset: 0

Author

N. J. A. Sloane, Dec 31 2003

Keywords

• nonn

Comments

Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=4.

References

• A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-5,8,-11,13,-15,16,-15,13,-11,8,-5,3,-1).

Programs

Mathematica

CoefficientList[Series[(1+x)*(1+x^3)*(1+x^5)*(1+x^7)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)), {x, 0, 70}], x] (* Jinyuan Wang, Mar 10 2020 *)

PARI

Vec((1+x)*(1+x^3)*(1+x^5)*(1+x^7)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8))+ O(x^100)) \\ Michel Marcus, Mar 19 2014

Formula

G.f.: (x^4-x^3+x^2-x+1)*(x^6-x^5+x^4-x^3+x^2-x+1) / ( (1+x+x^2)*(x^4+1)*(x^2+1)^2*(x-1)^4 ). - R. J. Mathar, Dec 18 2014

a(n)= +3*a(n-1) -5*a(n-2) +8*a(n-3) -11*a(n-4) +13*a(n-5) -15*a(n-6) +16*a(n-7) -15*a(n-8) +13*a(n-9) -11*a(n-10) +8*a(n-11) -5*a(n-12) +3*a(n-13) -a(n-14). - R. J. Mathar, Dec 18 2014