A008612 Molien series of 2-dimensional representation of SL(2,3).

1, 0, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 3, 3, 4, 4, 3, 5, 4, 4, 5, 5, 4, 6, 5, 5, 6, 6, 5, 7, 6, 6, 7, 7, 6, 8, 7, 7, 8, 8, 7, 9, 8, 8, 9, 9, 8, 10, 9, 9, 10, 10, 9, 11, 10, 10, 11, 11, 10, 12, 11, 11, 12, 12, 11, 13, 12, 12, 13, 13, 12, 14

Offset

0,7

Comments

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 2 ).

References

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))

William A. Stein, The modular forms database

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Formula

From Colin Barker, Jan 07 2014: (Start)

a(n) = a(n-2) + a(n-3) - a(n-5).

G.f.: (1-x^2+x^4) / ((1-x)^2*(1+x)*(1+x+x^2)). (End)

a(n) ~ n/6 (first difference is 6-periodic). - Ralf Stephan, Apr 29 2014

a(n) = 2*floor(n/3) -n/2 +(3+(-1)^n)/4. - Tani Akinari, Oct 23 2014

12*a(n) = 1 +2*n +3*(-1)^n +8*A057078(n). - R. J. Mathar, Jan 14 2021

Maple

(1+x^12)/(1-x^6)/(1-x^8); seq(coeff(series(%, x, 2*n+1), x, 2*n), n=0..100);

Mathematica

CoefficientList[Series[(1-x^2+x^4)/((1-x)^2*(1+x)*(1+x+x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Oct 23 2014 *)

Prog

(PARI) Vec((1-x^2+x^4)/((1-x)^2*(1+x)*(1+x+x^2)) + O(x^100)) \\ Colin Barker, Jan 07 2014
(Magma) [2*Floor(n/3)-n/2+(3+(-1)^n)/4: n in [0..100]]; // Vincenzo Librandi, Oct 23 2014
(Sage)
def A008612_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x^6)/((1-x^3)*(1-x^4)) ).list()
A008612_list(100) # G. C. Greubel, Feb 06 2020

Crossrefs

Sequence in context: A074746 A133188 A261036 * A029320 A363992 A187450

Adjacent sequences: A008609 A008610 A008611 * A008613 A008614 A008615

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved