OFFSET
0,5
COMMENTS
A_8 = SL_2(4) and acts on F_2[x_1, ..., x_4]. There are two copies of A_5 inside A_8. This is the Poincaré series (or Molien series) for the subgroup A_5 acting on F_2[x_1, ..., x_4] by tensoring over F_2 from the action of S_5 on Z^4 where Z^4 consists of those elements (n_1, ..., n_5) with Sum n_i = 0. That is, A_5 acts on the subring F_2[x_1 - x_5, x_2 - x_5, x_3 - x_5, x_4 - x_5] subset F_2[x_1, ..., x_5] by restriction to A_5 of the permutation S_5 action. See A089596 for the other A_5.
a(n) is the number of partitions of n into parts 2, 3, 4, 5, and 10 containing at most one part 10. - Joerg Arndt, Aug 15 2020
REFERENCES
A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 113.
H. Derksen and G. Kemper, Computational Invariant Theory, Springer, 2002; p. 130.
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-1,0,-1,-1,1,2,0,-1).
FORMULA
a(n) ~ 1/360*n^3 + 1/60*n^2. - Ralf Stephan, Apr 29 2014
G.f.: ( 1-x^2-x^6+x^4+x^8 ) / ( (1+x+x^2)*(1+x+x^3+x^2+x^4)*(1+x)^2*(x-1)^4 ). - R. J. Mathar, Dec 18 2014
Euler transform of length 20 sequence [ 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Jul 19 2015
a(n) = - a(-4-n) for all n in Z. - Michael Somos, Jul 19 2015
EXAMPLE
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + ...
PROG
(PARI) {a(n) = (n^3 + 6*n^2 + 96*n - 45*(n%2)*(n+2) - 9*(n%15==11)) \ 360 + 1}; /* Michael Somos, Jul 19 2015 */
(PARI) {a(n) = my(s=1); if( n<0, n = -4-n; s = -1); s * polcoeff( (1 + x^10) / ((1 - x^2) * (1 - x^3) * (1 - x^4) * (1 - x^5)) + x * O(x^n), n)}; /* Michael Somos, Jul 19 2015 */
(PARI) Vec((1+x^10)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) + O(x^80)) \\ Michel Marcus, Jul 19 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 02 2004
STATUS
approved