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A090492 G.f.: (1+x^10)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)). 1

%I #28 Dec 17 2023 15:08:31

%S 1,0,1,1,2,2,3,3,5,5,8,7,11,11,15,16,20,21,27,28,35,36,44,46,55,58,67,

%T 71,82,86,99,103,117,123,138,145,161,169,187,196,216,225,247,258,281,

%U 294,318,332,359,374,403,419,450,468,501,521,555,577,614,637,677,701,743,770,814

%N G.f.: (1+x^10)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).

%C 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.

%C 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

%D A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 113.

%D H. Derksen and G. Kemper, Computational Invariant Theory, Springer, 2002; p. 130.

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,-1,-1,0,-1,-1,1,2,0,-1).

%F a(n) ~ 1/360*n^3 + 1/60*n^2. - _Ralf Stephan_, Apr 29 2014

%F 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

%F 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

%F a(n) = - a(-4-n) for all n in Z. - _Michael Somos_, Jul 19 2015

%e 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 + ...

%o (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 */

%o (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 */

%o (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

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Feb 02 2004

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)