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)).

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 20, 26, 32, 38, 47, 58, 69, 81, 96, 114, 133, 153, 177, 206, 236, 267, 304, 346, 390, 437, 490, 550, 613, 679, 753, 835, 921, 1011, 1111, 1221, 1335, 1455, 1586, 1728, 1877, 2032, 2200, 2382, 2571, 2768, 2980, 3207, 3443, 3689, 3952

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=5.

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, 14, -18, 21, -23, 24, -24, 23, -21, 18, -14, 11, -8, 5, -3, 1).

Crossrefs

Cf. A006950, A000933, A089597.

Programs

Mathematica

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 *)
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 *)

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

Formula

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