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A014670 G.f.: (1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)). 7

%I #26 Jun 28 2023 21:06:19

%S 1,1,1,2,3,4,5,7,10,13,16,20,26,32,38,47,58,69,81,96,114,133,153,177,

%T 206,236,267,304,346,390,437,490,550,613,679,753,835,921,1011,1111,

%U 1221,1335,1455,1586,1728,1877,2032,2200,2382,2571,2768,2980,3207,3443,3689,3952

%N G.f.: (1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)).

%C 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=5.

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

%H Seiichi Manyama, <a href="/A014670/b014670.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (3, -5, 8, -11, 14, -18, 21, -23, 24, -24, 23, -21, 18, -14, 11, -8, 5, -3, 1).

%F G.f.: -(x^2-x+1) *(x^6-x^5+x^4-x^3+x^2-x+1) *(x^6-x^3+1) / ( (x^4+x^3+x^2+x+1) *(1+x+x^2) *(x^4+1) *(x^2+1)^2 *(x-1)^5 ). - _R. J. Mathar_, Dec 18 2014

%t CoefficientList[Series[(1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)), {x, 0, 50}], x] (* _Jinyuan Wang_, Mar 10 2020 *)

%t LinearRecurrence[{3,-5,8,-11,14,-18,21,-23,24,-24,23,-21,18,-14,11,-8,5,-3,1},{1,1,1,2,3,4,5,7,10,13,16,20,26,32,38,47,58,69,81},60] (* _Harvey P. Dale_, Mar 28 2023 *)

%o (PARI) Vec((1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10))+ O(x^100)) \\ _Michel Marcus_, Mar 18 2014

%Y Cf. A006950, A000933, A089597.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Dec 31 2003

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